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Elasticity/Tensors

Tensors in Solid Mechanics
A sound understanding of tensors and tensor operation is essential if you want to read and understand modern papers on solid mechanics and finite element modeling of complex material behavior. This brief introduction gives you an overview of tensors and tensor notation. For more details you can read  A Brief on Tensor Analysis by J. G. Simmonds, the appendix on vector and tensor notation from Dynamics of Polymeric Liquids - Volume 1 by R. B. Bird, R. C. Armstrong, and O. Hassager, and the monograph by R. M. Brannon. An introduction to tensors in continuum mechanics can be found in An Introduction to Continuum Mechanics by M. E. Gurtin. Most of the material in this page is based on these sources.

Notation
The following notation is usually used in the literature:
 * $$\begin{align}

s & = ~\text{scalar (lightface italic small)} \\ \mathbf{v} & = ~\text{vector (boldface roman small)} \\ \boldsymbol{\sigma} & = ~\text{second-order tensor (boldface Greek)} \\ \boldsymbol{A} & = ~\text{third-order tensor (boldface italic capital)} \\ \boldsymbol{\mathsf{A}} & = ~\text{fourth-order tensor (sans-serif capital)} \end{align}$$

Motivation
A  force $$\mathbf{f}\,$$ has a magnitude and a direction, can be added to another force, be multiplied by a scalar and so on. These properties make the force $$\mathbf{f}\,$$ a  vector.

Similarly, the  displacement $$\mathbf{u}$$ is a  vector because it can be added to other displacements and satisfies the other properties of a vector.

However, a force cannot be added to a displacement to yield a physically meaningful quantity. So the physical spaces that these two quantities lie on must be different.

Recall that a constant force $$\mathbf{f}$$ moving through a displacement $$\mathbf{u}\,$$ does $$\mathbf{f}\bullet\mathbf{u}$$ units of work. How do we compute this product when the spaces of $$\mathbf{f}\,$$ and $$\mathbf{u}\,$$ are different? If you try to compute the product on a graph, you will have to convert both quantities to a single basis and then compute the scalar product.

An alternative way of thinking about the operation $$\mathbf{f}\bullet\mathbf{u}$$ is to think of $$\mathbf{f}\,$$ as a  linear operator that acts on $$\mathbf{u}$$ to produce a scalar quantity (work). In the notation of sets we can write

\mathbf{f}\bullet\mathbf{u} \equiv\mathbf{f} : \mathbf{u} \rightarrow \mathbb{R}^{}~. $$ A  first order tensor is a linear operator that sends vectors to scalars.

Next, assume that the force $$\mathbf{f}\,$$ acts at a point $$\mathbf{x}\,$$. The moment of the force about the origin is given by $$\mathbf{x}\times\mathbf{f}\,$$ which is a  vector. The vector product can be thought of as an  linear operation too. In this case the effect of the operator is to convert a vector into another vector.

A  second order tensor is a linear operator that sends vectors to vectors. According to Simmonds, "the name  tensor comes from elasticity theory where in a loaded elastic body the stress tensor acting on a unit vector normal to a plane through a point delivers the tension (i.e., the force per unit area) acting across the plane at that point."

Examples of second order tensors are the stress tensor, the deformation gradient tensor, the velocity gradient tensor, and so on.

Another type of tensor that we encounter frequently in mechanics is the  fourth order tensor that takes strains to stresses. In elasticity, this is the stiffness tensor.

A  fourth order tensor is a linear operator that sends second order tensors to second order tensors.

Tensor algebra
A tensor $$\boldsymbol{A}\,$$ is a linear transformation from a vector space $$\mathcal{V}$$ to $$\mathcal{V}$$. Thus, we can write

\boldsymbol{A} : \mathbf{u} \in \mathcal{V} \rightarrow \mathbf{v \in \mathcal{V}}~. $$

More often, we use the following notation:

\mathbf{v} = \boldsymbol{A} \mathbf{u} \equiv \boldsymbol{A}(\mathbf{u}) \equiv \boldsymbol{A}\bullet\mathbf{u}~. $$

I have used the "dot" notation in this handout. None of the above notations is obviously superior to the others and each is used widely.

Addition of tensors
Let $$\boldsymbol{A}\,$$ and $$\boldsymbol{B}\,$$ be two tensors. Then the sum $$(\boldsymbol{A} + \boldsymbol{B})\,$$ is another tensor $$\boldsymbol{C}\,$$ defined by

\boldsymbol{C} = \boldsymbol{A} + \boldsymbol{B} \implies \boldsymbol{C}\bullet\mathbf{v} = (\boldsymbol{A} + \boldsymbol{B})\bullet\mathbf{v} = \boldsymbol{A}\bullet\mathbf{v} + \boldsymbol{B}\bullet\mathbf{v} ~. $$

Multiplication of a tensor by a scalar
Let $$\boldsymbol{A}\,$$ be a tensor and let $$\lambda\,$$ be a scalar. Then the product $$\boldsymbol{C} = \lambda \boldsymbol{A}\,$$ is a tensor defined by

\boldsymbol{C} = \lambda \boldsymbol{A} \implies \boldsymbol{C}\bullet\mathbf{v} = (\lambda \boldsymbol{A})\bullet\mathbf{v} = \lambda (\boldsymbol{A}\bullet\mathbf{v}) ~. $$

Zero tensor
The zero tensor $$\boldsymbol{\mathit{0}}\,$$ is the tensor which maps every vector $$\mathbf{v}\,$$ into the zero vector.

\boldsymbol{\mathit{0}}\bullet\mathbf{v} = \mathbf{0} ~. $$

Identity tensor
The identity tensor $$\boldsymbol{\mathit{I}}\,$$ takes every vector $$\mathbf{v}\,$$ into itself.

\boldsymbol{\mathit{I}}\bullet\mathbf{v} = \mathbf{v} ~. $$ The identity tensor is also often written as $$\boldsymbol{\mathit{1}}\,$$.

Product of two tensors
Let $$\boldsymbol{A}\,$$ and $$\boldsymbol{B}\,$$ be two tensors. Then the product $$\boldsymbol{C} = \boldsymbol{A}\bullet\boldsymbol{B}$$ is the tensor that is defined by

\boldsymbol{C} = \boldsymbol{A}\bullet\boldsymbol{B} \implies \boldsymbol{C}\bullet\mathbf{v} = (\boldsymbol{A}\bullet\boldsymbol{B})\bullet{\mathbf{v}} = \boldsymbol{A}\bullet(\boldsymbol{B}\bullet{\mathbf{v}}) ~. $$ In general $$\boldsymbol{A}\bullet\boldsymbol{B} \ne \boldsymbol{B}\bullet\boldsymbol{A}$$.

Transpose of a tensor
The transpose of a tensor $$\boldsymbol{A}\,$$ is the unique tensor $$\boldsymbol{A}^T\,$$ defined by

(\boldsymbol{A}\bullet\mathbf{u})\bullet\mathbf{v} = \mathbf{u}\bullet(\boldsymbol{A}^T\bullet\mathbf{v})~. $$ The following identities follow from the above definition:
 * $$\begin{align}

(\boldsymbol{A} + \boldsymbol{B})^T & = \boldsymbol{A}^T + \boldsymbol{B}^T ~, \\ (\boldsymbol{A}\bullet\boldsymbol{B})^T & = \boldsymbol{B}^T\bullet\boldsymbol{A}^T ~, \\ (\boldsymbol{A}^T)^T & = \boldsymbol{A} ~. \end{align}$$

Symmetric and skew tensors
A tensor $$\boldsymbol{A}\,$$ is symmetric if

\boldsymbol{A} = \boldsymbol{A}^T ~. $$ A tensor $$\boldsymbol{A}\,$$ is skew if

\boldsymbol{A} = -\boldsymbol{A}^T ~. $$

Every tensor $$\boldsymbol{A}\,$$ can be expressed uniquely as the sum of a symmetric tensor $$\boldsymbol{E}\,$$ (the symmetric part of $$\boldsymbol{A}\,$$) and a skew tensor $$\boldsymbol{W}\,$$ (the skew part of $$\boldsymbol{A}\,$$).

\boldsymbol{A} = \boldsymbol{E} + \boldsymbol{W} ~; \boldsymbol{E} = \cfrac{\boldsymbol{A} + \boldsymbol{A}^T}{2} ~; \boldsymbol{W} = \cfrac{\boldsymbol{A} - \boldsymbol{A}^T}{2} ~. $$

Tensor product of two vectors
The tensor (or dyadic) product $$\mathbf{a}\mathbf{b}\,$$ (also written $$\mathbf{a}\otimes\mathbf{b}\,$$) of two vectors $$\mathbf{a}\,$$ and $$\mathbf{b}\,$$ is a tensor that assigns to each vector $$\mathbf{v}\,$$ the vector $$(\mathbf{b}\bullet\mathbf{v})\mathbf{a}$$.

(\mathbf{a}\mathbf{b})\bullet\mathbf{v} = (\mathbf{a}\otimes\mathbf{b})\bullet\mathbf{v} = (\mathbf{b}\bullet\mathbf{v})\mathbf{a} ~. $$ Notice that all the above operations on tensors are remarkably similar to matrix operations.

Spectral theorem
The spectral theorem for tensors is widely used in mechanics. We will start off by definining eigenvalues and eigenvectors.

Eigenvalues and eigenvectors
Let $$\boldsymbol{S}$$ be a second order tensor. Let $$\lambda$$ be a scalar and $$\mathbf{n}$$ be a vector such that

\boldsymbol{S}\cdot\mathbf{n} = \lambda~\mathbf{n} $$ Then $$ \lambda $$ is called an  eigenvalue  of $$\boldsymbol{S}$$ and $$ \mathbf{n} $$ is an  eigenvector .

A second order tensor has three eigenvalues and three eigenvectors, since the space is three-dimensional. Some of the eigenvalues might be repeated. The number of times an eigenvalue is repeated is called multiplicity.

In mechanics, many second order tensors are symmetric and positive definite. Note the following important properties of such tensors:

 * 1) If $$\boldsymbol{S}$$ is positive definite, then $$\lambda > 0$$.
 * 2) If $$\boldsymbol{S}$$ is symmetric, the eigenvectors $$\mathbf{n}$$ are mutually orthogonal.

For more on eigenvalues and eigenvectors see Applied linear operators and spectral methods.

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Spectral theorem
Let $$\boldsymbol{S}$$ be a symmetric second-order tensor. Then
 * 1) the normalized eigenvectors $$\mathbf{n}_1, \mathbf{n}_2, \mathbf{n}_3$$ form an orthonormal basis.
 * 2) if $$ \lambda_1, \lambda_2, \lambda_3 $$ are the corresponding eigenvalues then $$ \boldsymbol{S} = \sum_{i=1}^3 \lambda_i \mathbf{n}_i \otimes \mathbf{n}_i $$.

This relation is called the spectral decomposition of $$\boldsymbol{S}$$.
 * }

Polar decomposition theorem
Let $$ \boldsymbol{F} $$ be second order tensor with $$ \det\boldsymbol{F} > 0 $$. Then
 * 1) there exist positive definite, symmetric tensors $$ \boldsymbol{U} $$,$$ \boldsymbol{V} $$ and a rotation (orthogonal) tensor $$ \boldsymbol{R} $$ such that $$ \boldsymbol{F} = \boldsymbol{R}\cdot \boldsymbol{U} = \boldsymbol{V} \cdot \boldsymbol{R} $$.
 * 2) also each of these decompositions is unique.

Principal invariants of a tensor
Let $$ \boldsymbol{S} $$ be a second order tensor. Then the determinant of $$ \boldsymbol{S} - \lambda~\boldsymbol{\mathit{I}} $$ can be expressed as

\det(\boldsymbol{S} - \lambda~\boldsymbol{\mathit{I}}) = -\lambda^3 + I_1(\boldsymbol{S})~\lambda^2 - I_2(\boldsymbol{S})~\lambda + I_3(\boldsymbol{S}) $$ The quantities $$ I_1, I_2, I_3\,$$ are called the principal invariants of $$\boldsymbol{S}$$. Expressions of the principal invariants are given below.

Note that $$ \lambda $$ is an eigenvalue of $$ \boldsymbol{S} $$ if and only if

\det(\boldsymbol{S} - \lambda~\boldsymbol{\mathit{I}}) = 0 $$ The resulting equations is called the characteristic equation and is usually written in expanded form as

\lambda^3 - I_1(\boldsymbol{S})~\lambda^2 + I_2(\boldsymbol{S})~\lambda -I_3(\boldsymbol{S}) = 0 $$

Cayley-Hamilton theorem
The Cayley-Hamilton theorem is a very useful result in continuum mechanics. It states that

Index notation
All the equations so far have made no mention of the coordinate system. When we use vectors and tensor in computations we have to express them in some coordinate system (basis) and use the components of the object in that basis for our computations.

Commonly used bases are the Cartesian coordinate frame, the cylindrical coordinate frame, and the spherical coordinate frame.

A  Cartesian coordinate frame consists of an orthonormal basis $$(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)\,$$ together with a point $$\mathbf{o}\,$$ called the origin. Since these vectors are mutually perpendicular, we have the following relations:
 * $$\begin{align}\text{(1)} \qquad

\mathbf{e}_1\bullet\mathbf{e}_1 & = 1 ~; \mathbf{e}_1\bullet\mathbf{e}_2 = 0 ~; \mathbf{e}_1\bullet\mathbf{e}_3 = 0 ~; \\ \mathbf{e}_2\bullet\mathbf{e}_1 & = 0 ~; \mathbf{e}_2\bullet\mathbf{e}_2 = 1 ~; \mathbf{e}_2\bullet\mathbf{e}_3 = 0 ~;\\ \mathbf{e}_3\bullet\mathbf{e}_1 & = 0 ~; \mathbf{e}_3\bullet\mathbf{e}_2 = 0 ~; \mathbf{e}_3\bullet\mathbf{e}_3 = 1 ~. \end{align}$$

Kronecker delta
To make the above relations more compact, we introduce the  Kronecker delta symbol

{   \delta_{ij} = \begin{cases} 1 & ~\rm{if}~ i = j~. \\                    0 & ~\rm{if}~ i \ne j ~. \end{cases} } $$ Then, instead of the nine equations in (1) we can write (in  index notation)

\mathbf{e}_i\bullet\mathbf{e}_j = \delta_{ij} ~. $$

Einstein summation convention
Recall that the vector $$\mathbf{u}\,$$ can be written as
 * $$\text{(2)} \qquad

\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 = \sum_{i=1}^3 u_i \mathbf{e}_i ~. $$

In index notation, equation (2) can be written as

{   \mathbf{u} = u_i \mathbf{e}_i~. } $$ This convention is called the  Einstein summation convention. If indices are repeated, we understand that to mean that there is a sum over the indices.

Components of a vector
We can write the Cartesian components of a vector $$\mathbf{u}\,$$ in the basis $$(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)\,$$ as

u_i = \mathbf{e}_i\bullet\mathbf{u} ~,i = 1, 2, 3~. $$

Components of a tensor
Similarly, the components $$A_{ij}\,$$ of a tensor $$\boldsymbol{A}\,$$ are defined by

{   A_{ij} = \mathbf{e}_i\bullet(\boldsymbol{A}\bullet\mathbf{e}_j)~. } $$ Using the definition of the tensor product, we can also write

\boldsymbol{A} = \sum_{i,j=1}^3 A_{ij} \mathbf{e}_i\mathbf{e}_j \equiv \sum_{i,j=1}^3 A_{ij} \mathbf{e}_i\otimes\mathbf{e}_j ~. $$ Using the summation convention,

{   \boldsymbol{A} =  A_{ij} \mathbf{e}_i\mathbf{e}_j \equiv A_{ij} \mathbf{e}_i\otimes\mathbf{e}_j~. } $$ In this case, the bases of the tensor are $$\{\mathbf{e}_i\otimes\mathbf{e}_j\}$$ and the components are $$A_{ij}\,$$.

Operation of a tensor on a vector
From the definition of the components of tensor $$\boldsymbol{A}\,$$, we can also see that (using the summation convention)

{   \mathbf{v} = \boldsymbol{A}\bullet\mathbf{u} \equiv v_i = A_{ij} u_j~. } $$

Dyadic product
Similarly, the dyadic product can be expressed as

{   (\mathbf{a}\mathbf{b})_{ij} \equiv (\mathbf{a}\otimes\mathbf{b})_{ij} = a_i b_j ~. } $$

Matrix notation
We can also write a tensor $$\boldsymbol{A}$$ in matrix notation as

\boldsymbol{A} = A_{ij}\mathbf{e}_i\mathbf{e}_j = A_{ij}\mathbf{e}_i\otimes\mathbf{e}_j \implies \mathbf{A} = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} ~. $$

Note that the Kronecker delta represents the components of the identity tensor in a Cartesian basis. Therefore, we can write

\boldsymbol{I} = \delta_{ij}\mathbf{e}_i\mathbf{e}_j = \delta_{ij}\mathbf{e}_i\otimes\mathbf{e}_j \implies \mathbf{I} = \begin{bmatrix} 1 & 0 & 0 \\                     0 & 1 & 0 \\                      0 & 0 & 1                       \end{bmatrix} ~. $$

Tensor inner product
The  inner product $$\boldsymbol{A} : \boldsymbol{B}\,$$ of two tensors $$\boldsymbol{A}\,$$ and $$\boldsymbol{B}\,$$ is an operation that generates a scalar. We define (summation implied)

{   \boldsymbol{A} : \boldsymbol{B} = A_{ij} B_{ij} ~. } $$ The inner product can also be expressed using the trace :

{   \boldsymbol{A} : \boldsymbol{B} = Tr(\boldsymbol{A^T} \bullet \boldsymbol{B}) ~. } $$

Proof using the definition of the trace below :

{   Tr(\boldsymbol{A^T} \bullet \boldsymbol{B}) = \boldsymbol{I}:(\boldsymbol{A^T} \bullet \boldsymbol{B})=\delta_{ij}\mathbf{e}_i\otimes\mathbf{e}_j :(A_{lk}\mathbf{e}_k \otimes \mathbf{e}_l \bullet B_{mn}\mathbf{e}_m\otimes \mathbf{e}_n) = \delta_{ij}\mathbf{e}_i\otimes\mathbf{e}_j : (A_{mk}B_{mn}\mathbf{e}_k \otimes \mathbf{e}_n) =  }$$

{   A_{mk}B_{mn}\delta_{ij}\delta_{in}\delta_{jk}= A_{mk}B_{mi}\delta_{ij}\delta_{jk} =A_{mk}B_{mj}\delta_{jk} = A_{mj}B_{mj} =A:B } $$

Trace of a tensor
The trace of a tensor is the scalar given by

\text{Tr}(\boldsymbol{A}) = \boldsymbol{I}:\boldsymbol{A} = \delta_{ij}\mathbf{e}_i\otimes\mathbf{e}_j:A_{mn}\mathbf{e}_m\otimes\mathbf{e}_n = \delta_{ij}\delta_{im}\delta_{jn}A_{mn} = A_{ii} $$

The trace of an N x N-matrix is the sum of the components on the downward-sloping diagonal.

Magnitude of a tensor
The  magnitude of a tensor $$\boldsymbol{A}\,$$ is defined by

\Vert \boldsymbol{A} \Vert = \sqrt{\boldsymbol{A}:\boldsymbol{A}} \equiv \sqrt{A_{ij}A_{ij}} ~. $$

Tensor product of a tensor with a vector
Another tensor operation that is often seen is the tensor product of a tensor with a vector. Let $$\boldsymbol{A}\,$$ be a tensor and let $$\mathbf{v}\,$$ be a vector. Then the tensor cross product gives a tensor $$\boldsymbol{C}\,$$ defined by

{   \boldsymbol{C} = \boldsymbol{A}\times\mathbf{v} \implies C_{ij} = e_{klj} A_{ik} v_{l} ~. } $$

Permutation symbol
The permutation symbol $$e_{ijk}\,$$ is defined as

{   e_{ijk} = \begin{cases} 1 & ~\text{if}~ ijk = 123, 231, ~\text{or}~ 312 \\ -1 & ~\text{if}~ ijk = 321, 132, ~\text{or}~ 213 \\ 0 & ~\text{if any two indices are alike} \end{cases} } $$

Identities in tensor algebra
Let $$\boldsymbol{A}$$, $$\boldsymbol{B}$$ and $$\boldsymbol{C}$$ be three second order tensors. Then

\boldsymbol{A}:(\boldsymbol{B}\cdot\boldsymbol{C}) = (\boldsymbol{C}\cdot\boldsymbol{A}^T):\boldsymbol{B}^T = (\boldsymbol{B}^T\cdot\boldsymbol{A}):\boldsymbol{C} $$

 Proof:  It is easiest to show these relations by using index notation with respect to an orthonormal basis. Then we can write

\boldsymbol{A}:(\boldsymbol{B}\cdot\boldsymbol{C}) \equiv A_{ij} (B_{ik}~C_{kj}) = C_{kj}~A^T_{ji}~B^T_{ki} \equiv (\boldsymbol{C}\cdot\boldsymbol{A}^T):\boldsymbol{B}^T $$ Similarly,

\boldsymbol{A}:(\boldsymbol{B}\cdot\boldsymbol{C}) \equiv A_{ij} (B_{ik}~C_{kj}) = B^T_{ki}~A_{ij}~C_{kj} \equiv (\boldsymbol{B}^T\cdot\boldsymbol{A}):\boldsymbol{C} $$

Tensor calculus
Recall that the vector differential operator (with respect to a Cartesian basis) is defined as

\boldsymbol{\nabla}{} = \cfrac{\partial }{\partial x_1}\mathbf{e}_1+\cfrac{\partial }{\partial x_2}\mathbf{e}_2+\cfrac{\partial }{\partial x_3}\mathbf{e}_3 \equiv \cfrac{\partial }{\partial x_i}\mathbf{e}_i ~. $$ In this section we summarize some operations of $$\boldsymbol{\nabla}{}$$ on vectors and tensors.

The gradient of a vector field
The dyadic product $$\boldsymbol{\nabla}{\mathbf{v}}\,$$ (or $$\boldsymbol{\nabla}{}\otimes\mathbf{v}$$) is called the  gradient of the vector field $$\mathbf{v}\,$$. Therefore, the quantity $$\boldsymbol{\nabla}{\mathbf{v}}$$ is a  tensor given by

{   \boldsymbol{\nabla}{\mathbf{v}} = \sum_i\sum_j \cfrac{\partial v_j}{\partial x_i} \mathbf{e}_i \mathbf{e}_j \equiv v_{j,i} \mathbf{e}_i \mathbf{e}_j ~. } $$ In the alternative dyadic notation,

{   \boldsymbol{\nabla}{\mathbf{v}} \equiv \boldsymbol{\nabla}{}\otimes\mathbf{v} = \sum_i\sum_j \cfrac{\partial v_j}{\partial x_i} \mathbf{e}_i\otimes\mathbf{e}_j \equiv v_{j,i} \mathbf{e}_i\otimes\mathbf{e}_j ~. } $$ 'Warning: Some authors define the $$ij$$ component of $$\boldsymbol{\nabla}{\mathbf{v}}$$ as $$\partial v_i/\partial x_j = v_{i,j}$$.

The divergence of a tensor field
Let $$\boldsymbol{A}\,$$ be a tensor field. Then the  divergence of the tensor field is a  vector $$\boldsymbol{\nabla}\bullet{\boldsymbol{A}}$$ given by

{   \boldsymbol{\nabla}\bullet{\boldsymbol{A}} = \sum_j \left[\sum_i \cfrac{\partial A_{ij}}{\partial x_i}\right] \mathbf{e}_j \equiv \cfrac{\partial A_{ij}}{\partial x_i} \mathbf{e}_j = A_{ij,i} \mathbf{e}_j~. } $$ To fix the definition of divergence of a general tensor field (possibly of higher order than 2), we use the relation

(\boldsymbol{\nabla}\bullet{\boldsymbol{A}})\bullet\mathbf{c} = \boldsymbol{\nabla}\bullet(\boldsymbol{A}\bullet\mathbf{c}) $$ where $$\mathbf{c}$$ is an arbitrary constant vector.

The Laplacian of a vector field
The  Laplacian of a  vector field is given by

{   \nabla^2{\mathbf{v}} = \boldsymbol{\nabla}\bullet{\boldsymbol{\nabla}{\mathbf{v}}} = \sum_j \left[\sum_i \cfrac{\partial^2 v_j}{\partial x_i^2}\right] \mathbf{e}_j \equiv v_{j,ii} \mathbf{e}_j ~. } $$

Tensor Identities
Some important identities involving tensors are:
 * 1) $$\boldsymbol{\nabla}\bullet{\boldsymbol{\nabla}{\mathbf{v}}} = \boldsymbol{\nabla}{(\boldsymbol{\nabla}\bullet{\mathbf{v}})} - \boldsymbol{\nabla}\times{(\boldsymbol{\nabla}\times{\mathbf{v}})}$$.
 * 2) $$\mathbf{v}\bullet\boldsymbol{\nabla}{\mathbf{v}} = \frac{1}{2}\boldsymbol{\nabla}{(\mathbf{v}\bullet\mathbf{v})} - \mathbf{v}\times(\boldsymbol{\nabla}\times{\mathbf{v})}$$.
 * 3) $$\boldsymbol{\nabla}\bullet{(\mathbf{v}\otimes\mathbf{w})} = \mathbf{v}\bullet\boldsymbol{\nabla}{\mathbf{w}} + \mathbf{w}(\boldsymbol{\nabla}\bullet{\mathbf{v}})$$.
 * 4) $$\boldsymbol{\nabla}\bullet{(\varphi\boldsymbol{A})} = \boldsymbol{\nabla}{\varphi}\bullet\boldsymbol{A} + \varphi\boldsymbol{\nabla}\bullet{\boldsymbol{A}}$$.
 * 5) $$\boldsymbol{\nabla}{(\mathbf{v}\bullet\mathbf{w})} = (\boldsymbol{\nabla}{\mathbf{v}})\bullet\mathbf{w} + (\boldsymbol{\nabla}{\mathbf{w}})\bullet\mathbf{v}$$.
 * 6) $$\boldsymbol{\nabla}\bullet{(\boldsymbol{A}\bullet\mathbf{w})} = (\boldsymbol{\nabla}\bullet{\boldsymbol{A}})\bullet\mathbf{w} + \boldsymbol{A}^T:(\boldsymbol{\nabla}{\mathbf{w}})$$.

Integral theorems
The following integral theorems are useful in continuum mechanics and finite elements.

The Gauss divergence theorem
If $$\Omega$$ is a region in space enclosed by a surface $$\Gamma\,$$ and $$\boldsymbol{A}\,$$ is a tensor field, then

{   \int_{\Omega} \boldsymbol{\nabla}\bullet{\boldsymbol{A}} ~dV = \int_{\Gamma} \mathbf{n}\bullet\boldsymbol{A} ~dA } $$ where $$\mathbf{n}\,$$ is the unit outward normal to the surface.

The Stokes curl theorem
If $$\Gamma\,$$ is a surface bounded by a closed curve $$\mathcal{C}$$, then

\int_{\Gamma} \mathbf{n}\bullet(\boldsymbol{\nabla}\times{\boldsymbol{A})}~dA = \oint_{\mathcal{C}} \mathbf{t}\bullet\boldsymbol{A}~ ds $$ where $$\boldsymbol{A}\,$$ is a tensor field, $$\mathbf{n}\,$$ is the unit normal vector to $$\Gamma\,$$ in the direction of a right-handed screw motion along $$\mathcal{C}$$, and $$\mathbf{t}\,$$ is a unit tangential vector in the direction of integration along $$\mathcal{C}$$.

The Leibniz formula
Let $$\Omega$$ be a closed moving region of space enclosed by a surface $$\Gamma\,$$. Let the velocity of any surface element be $$\mathbf{v}\,$$. Then if $$\boldsymbol{A}(\mathbf{x},t)\,$$ is a tensor function of position and time,

\cfrac{\partial }{\partial t} \int_{\Omega} \boldsymbol{A}~dV = \int_{\Omega} \cfrac{\partial \boldsymbol{A}}{\partial t}~dV + \int_{\Gamma} \boldsymbol{A}(\mathbf{v}\bullet\mathbf{n})~dA $$ where $$\mathbf{n}\,$$ is the outward unit normal to the surface $$\Gamma\,$$.

Directional derivatives
We often have to find the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors. The directional directive provides a systematic way of finding these derivatives.

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Derivatives of scalar valued functions of vectors
Let $$f(\mathbf{v})$$ be a real valued function of the vector $$\mathbf{v}$$. Then the derivative of $$f(\mathbf{v})$$ with respect to $$\mathbf{v}$$ (or at $$\mathbf{v}$$) in the direction $$\mathbf{u}$$ is the  vector defined as

\frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}] = \left[\frac{\partial }{\partial \alpha}~f(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0} $$ for all vectors $$\mathbf{u}$$.

Properties:

1) If $$f(\mathbf{v}) = f_1(\mathbf{v}) + f_2(\mathbf{v})$$ then $$  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}} + \frac{\partial f_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}  $$

2) If $$f(\mathbf{v}) = f_1(\mathbf{v})~ f_2(\mathbf{v})$$ then $$  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial f_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)~f_2(\mathbf{v}) + f_1(\mathbf{v})~\left(\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right) $$

3) If $$f(\mathbf{v}) = f_1(f_2(\mathbf{v}))$$ then $$  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial f_1}{\partial f_2}~\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u}  $$

Derivatives of vector valued functions of vectors
Let $$\mathbf{f}(\mathbf{v})$$ be a vector valued function of the vector $$\mathbf{v}$$. Then the derivative of $$\mathbf{f}(\mathbf{v})$$ with respect to $$\mathbf{v}$$ (or at $$\mathbf{v}$$) in the direction $$\mathbf{u}$$ is the  second order tensor defined as

\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}] = \left[\frac{\partial }{\partial \alpha}~\mathbf{f}(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0} $$ for all vectors $$\mathbf{u}$$.

Properties:

1) If $$\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) + \mathbf{f}_2(\mathbf{v})$$ then $$  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}} + \frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}  $$

2) If $$\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v})\times\mathbf{f}_2(\mathbf{v})$$ then $$  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf{f}_2(\mathbf{v}) + \mathbf{f}_1(\mathbf{v})\times\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right) $$

3) If $$\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{f}_2(\mathbf{v}))$$ then $$  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} =  \frac{\partial \mathbf{f}_1}{\partial \mathbf{f}_2}\cdot\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right) $$

Derivatives of scalar valued functions of tensors
Let $$f(\boldsymbol{S})$$ be a real valued function of the second order tensor $$\boldsymbol{S}$$. Then the derivative of $$f(\boldsymbol{S})$$ with respect to $$\boldsymbol{S}$$ (or at $$\boldsymbol{S}$$) in the direction $$\boldsymbol{T}$$ is the  second order tensor defined as

\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = Df(\boldsymbol{S})[\boldsymbol{T}] = \left[\frac{\partial }{\partial \alpha}~f(\boldsymbol{S} + \alpha~\boldsymbol{T})\right]_{\alpha = 0} $$ for all second order tensors $$\boldsymbol{T}$$.

Properties:

1) If $$f(\boldsymbol{S}) = f_1(\boldsymbol{S}) + f_2(\boldsymbol{S})$$ then $$ \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}} + \frac{\partial f_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} $$

2) If $$f(\boldsymbol{S}) = f_1(\boldsymbol{S})~ f_2(\boldsymbol{S})$$ then $$ \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)~f_2(\boldsymbol{S}) + f_1(\boldsymbol{S})~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) $$

3) If $$f(\boldsymbol{S}) = f_1(f_2(\boldsymbol{S}))$$ then $$ \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial f_2}~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) $$

Derivatives of tensor valued functions of tensors
Let $$\boldsymbol{F}(\boldsymbol{S})$$ be a second order tensor valued function of the second order tensor $$\boldsymbol{S}$$. Then the derivative of $$\boldsymbol{F}(\boldsymbol{S})$$ with respect to $$\boldsymbol{S}$$ (or at $$\boldsymbol{S}$$) in the direction $$\boldsymbol{T}$$ is the  fourth order tensor defined as

\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = D\boldsymbol{F}(\boldsymbol{S})[\boldsymbol{T}] = \left[\frac{\partial }{\partial \alpha}~\boldsymbol{F}(\boldsymbol{S} + \alpha~\boldsymbol{T})\right]_{\alpha = 0} $$ for all second order tensors $$\boldsymbol{T}$$.

Properties:

1) If $$\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S}) + \boldsymbol{F}_2(\boldsymbol{S})$$ then $$ \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}} + \frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} $$

2) If $$\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})$$ then $$ \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2(\boldsymbol{S}) + \boldsymbol{F}_1(\boldsymbol{S})\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) $$

3) If $$\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{F}_2(\boldsymbol{S}))$$ then $$ \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) $$

3) If $$f(\boldsymbol{S}) = f_1(\boldsymbol{F}_2(\boldsymbol{S}))$$ then $$ \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) $$

Derivative of the determinant of a tensor
Proof: Let $$\boldsymbol{A}$$ be a second order tensor and let $$f(\boldsymbol{A}) = \det(\boldsymbol{A})$$. Then, from the definition of the derivative of a scalar valued function of a tensor, we have

\begin{align} \frac{\partial f}{\partial \boldsymbol{A}}:\boldsymbol{T} & = \left.\cfrac{d}{d\alpha} \det(\boldsymbol{A} + \alpha~\boldsymbol{T}) \right|_{\alpha=0} \\ & = \left.\cfrac{d}{d\alpha} \det\left[\alpha~\boldsymbol{A}\left(\cfrac{1}{\alpha}~\boldsymbol{\mathit{1}} + \boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right) \right] \right|_{\alpha=0} \\ & = \left.\cfrac{d}{d\alpha} \left[\alpha^3~\det(\boldsymbol{A})~ \det\left(\cfrac{1}{\alpha}~\boldsymbol{\mathit{1}} + \boldsymbol{A}^{-1}\cdot\boldsymbol{T}\right)\right] \right|_{\alpha=0} ~. \end{align} $$ Recall that we can expand the determinant of a tensor in the form of a characteristic equation in terms of the invariants $$I_1,I_2,I_3$$ using (note the sign of $$\lambda$$)

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) = \lambda^3 + I_1(\boldsymbol{A})~\lambda^2 + I_2(\boldsymbol{A})~\lambda + I_3(\boldsymbol{A}) ~. $$

Using this expansion we can write

\begin{align} \frac{\partial f}{\partial \boldsymbol{A}}:\boldsymbol{T} & = \left.\cfrac{d}{d\alpha} \left[\alpha^3~\det(\boldsymbol{A})~ \left(\cfrac{1}{\alpha^3} + I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\cfrac{1}{\alpha^2} +      I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\cfrac{1}{\alpha} + I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})\right) \right] \right|_{\alpha=0} \\ & = \left.\det(\boldsymbol{A})~\cfrac{d}{d\alpha} \left[ 1 + I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha + I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^2 + I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^3 \right] \right|_{\alpha=0} \\ & = \left.\det(\boldsymbol{A})~\left[I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) + 2~I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha + 3~I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^2 \right] \right|_{\alpha=0} \\ & = \det(\boldsymbol{A})~I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) ~. \end{align} $$ Recall that the invariant $$I_1$$ is given by

I_1(\boldsymbol{A}) = \text{tr}{\boldsymbol{A}} ~. $$ Hence,

\frac{\partial f}{\partial \boldsymbol{A}}:\boldsymbol{T} = \det(\boldsymbol{A})~\text{tr}(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) = \det(\boldsymbol{A})~[\boldsymbol{A}^{-1}]^T : \boldsymbol{T} ~. $$ Invoking the arbitrariness of $$\boldsymbol{T}$$ we then have

\frac{\partial f}{\partial \boldsymbol{A}} = \det(\boldsymbol{A})~[\boldsymbol{A}^{-1}]^T ~. $$

Derivatives of the invariants of a tensor
Proof: From the derivative of the determinant we know that

\frac{\partial I_3}{\partial \boldsymbol{A}} = \det(\boldsymbol{A})~[\boldsymbol{A}^{-1}]^T ~. $$

For the derivatives of the other two invariants, let us go back to the characteristic equation

\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) = \lambda^3 + I_1(\boldsymbol{A})~\lambda^2 + I_2(\boldsymbol{A})~\lambda + I_3(\boldsymbol{A}) ~. $$ Using the same approach as for the determinant of a tensor, we can show that

\frac{\partial }{\partial \boldsymbol{A}}\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) = \det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~[(\lambda~\boldsymbol{\mathit{1}}+\boldsymbol{A})^{-1}]^T ~. $$ Now the left hand side can be expanded as

\begin{align} \frac{\partial }{\partial \boldsymbol{A}}\det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A}) & = \frac{\partial }{\partial \boldsymbol{A}}\left[ \lambda^3 + I_1(\boldsymbol{A})~\lambda^2 + I_2(\boldsymbol{A})~\lambda + I_3(\boldsymbol{A}) \right] \\ & =     \frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_3}{\partial \boldsymbol{A}}~. \end{align} $$ Hence

\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_3}{\partial \boldsymbol{A}} = \det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~[(\lambda~\boldsymbol{\mathit{1}}+\boldsymbol{A})^{-1}]^T $$ or,

(\lambda~\boldsymbol{\mathit{1}}+\boldsymbol{A})^T\cdot\left[ \frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_3}{\partial \boldsymbol{A}}\right] = \det(\lambda~\boldsymbol{\mathit{1}} + \boldsymbol{A})~\boldsymbol{\mathit{1}} ~. $$ Expanding the right hand side and separating terms on the left hand side gives

(\lambda~\boldsymbol{\mathit{1}} +\boldsymbol{A}^T)\cdot\left[ \frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_3}{\partial \boldsymbol{A}}\right] = \left[\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right] \boldsymbol{\mathit{1}} $$ or,

\begin{align} \left[\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^3 \right.& \left.+ \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_3}{\partial \boldsymbol{A}}~\lambda\right]\boldsymbol{\mathit{1}} + \boldsymbol{A}^T\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} \\ & =      \left[\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right] \boldsymbol{\mathit{1}} ~. \end{align} $$ If we define $$I_0 := 1$$ and $$I_4 := 0$$, we can write the above as

\begin{align} \left[\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^3 \right.& \left.+ \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda^2 + \frac{\partial I_3}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_4}{\partial \boldsymbol{A}}\right]\boldsymbol{\mathit{1}} + \boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}~\lambda^3 + \boldsymbol{A}^T\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 + \boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda + \boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} \\ &=      \left[I_0~\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right] \boldsymbol{\mathit{1}} ~. \end{align} $$ Collecting terms containing various powers of $$\lambda$$, we get

\begin{align} \lambda^3&\left(I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -                   \boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}\right) + \lambda^2\left(I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -                   \boldsymbol{A}^T\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}\right) + \\ &\qquad \qquad\lambda\left(I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -                   \boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}\right) + \left(I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -                   \boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}}\right)  = 0 ~. \end{align} $$ Then, invoking the arbitrariness of $$\lambda$$, we have

\begin{align} I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}} & = 0 \\ I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}} & = 0 \\ I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} & = 0 ~. \end{align} $$ This implies that

\begin{align} \frac{\partial I_1}{\partial \boldsymbol{A}} &= \boldsymbol{\mathit{1}} \\ \frac{\partial I_2}{\partial \boldsymbol{A}} & = I_1~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\\ \frac{\partial I_3}{\partial \boldsymbol{A}} & = I_2~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T) = (\boldsymbol{A}^2 - I_1~\boldsymbol{A} + I_2~\boldsymbol{\mathit{1}})^T \end{align} $$

Derivative of the identity tensor
Let $$\boldsymbol{\mathit{1}}$$ be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor $$\boldsymbol{A}$$ is given by

\frac{\partial \boldsymbol{\mathit{1}}}{\partial \boldsymbol{A}}:\boldsymbol{T} = \boldsymbol{\mathsf{0}}:\boldsymbol{T} = \boldsymbol{\mathit{0}} $$ This is because $$\boldsymbol{\mathit{1}}$$ is independent of $$\boldsymbol{A}$$.

Derivative of a tensor with respect to itself
Let $$\boldsymbol{A}$$ be a second order tensor. Then

\frac{\partial \boldsymbol{A}}{\partial \boldsymbol{A}}:\boldsymbol{T} = \left[\frac{\partial }{\partial \alpha} (\boldsymbol{A} + \alpha~\boldsymbol{T})\right]_{\alpha = 0} = \boldsymbol{T} = \boldsymbol{\mathsf{I}}:\boldsymbol{T} $$ Therefore,

\frac{\partial \boldsymbol{A}}{\partial \boldsymbol{A}} = \boldsymbol{\mathsf{I}} $$ Here $$\boldsymbol{\mathsf{I}}$$ is the fourth order identity tensor. In index notation with respect to an orthonormal basis

\boldsymbol{\mathsf{I}} = \delta_{ik}~\delta_{jl}~\mathbf{e}_i\otimes\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_l $$

This result implies that

\frac{\partial \boldsymbol{A}^T}{\partial \boldsymbol{A}}:\boldsymbol{T} = \boldsymbol{\mathsf{I}}^T:\boldsymbol{T} = \boldsymbol{T}^T $$ where

\boldsymbol{\mathsf{I}}^T = \delta_{jk}~\delta_{il}~\mathbf{e}_i\otimes\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_l $$

Therefore, if the tensor $$\boldsymbol{A}$$ is symmetric, then the derivative is also symmetric and we get

\frac{\partial \boldsymbol{A}}{\partial \boldsymbol{A}} = \frac{\partial \frac{1}{2}(\boldsymbol{A}+\boldsymbol{A}^T)}{\partial \boldsymbol{A}} = \frac{1}{2}~(\boldsymbol{\mathsf{I}} + \boldsymbol{\mathsf{I}}^T) = \boldsymbol{\mathsf{I}}^{(s)} $$ where the symmetric fourth order identity tensor is

\boldsymbol{\mathsf{I}}^{(s)} = \frac{1}{2}~(\delta_{ik}~\delta_{jl} + \delta_{il}~\delta_{jk}) ~\mathbf{e}_i\otimes\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_l $$

Derivative of the inverse of a tensor
 Proof: Recall that

\frac{\partial \boldsymbol{\mathit{1}}}{\partial \boldsymbol{A}}:\boldsymbol{T} = \boldsymbol{\mathit{0}} $$ Since $$\boldsymbol{A}^{-1}\cdot\boldsymbol{A} = \boldsymbol{\mathit{1}}$$, we can write

\frac{\partial }{\partial \boldsymbol{A}}(\boldsymbol{A}^{-1}\cdot\boldsymbol{A}):\boldsymbol{T} = \boldsymbol{\mathit{0}} $$ Using the product rule for second order tensors

\frac{\partial }{\partial \boldsymbol{S}}[\boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})]:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2 + \boldsymbol{F}_1\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T}\right) $$ we get

\frac{\partial }{\partial \boldsymbol{A}}(\boldsymbol{A}^{-1}\cdot\boldsymbol{A}):\boldsymbol{T} = \left(\frac{\partial \boldsymbol{A}^{-1}}{\partial \boldsymbol{A}}:\boldsymbol{T}\right)\cdot\boldsymbol{A} + \boldsymbol{A}^{-1}\cdot\left(\frac{\partial \boldsymbol{A}}{\partial \boldsymbol{A}}:\boldsymbol{T}\right) = \boldsymbol{\mathit{0}} $$ or,

\left(\frac{\partial \boldsymbol{A}^{-1}}{\partial \boldsymbol{A}}:\boldsymbol{T}\right)\cdot\boldsymbol{A} = - \boldsymbol{A}^{-1}\cdot\boldsymbol{T} $$ Therefore,

\frac{\partial }{\partial \boldsymbol{A}} \left(\boldsymbol{A}^{-1}\right) : \boldsymbol{T} = - \boldsymbol{A}^{-1}\cdot\boldsymbol{T}\cdot\boldsymbol{A}^{-1} $$

Remarks
The boldface notation that I've used is called the  Gibbs notation. The index notation that I have used is also called  Cartesian tensor notation.

title: Elasticity/Tensors


text: File:HWconsole.jpg

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: File:HWconsole.jpg


text:
C Programming/Basic Output

Objective

 * Learn how to output text to the console.

Lesson
Output is information that the program tells the user. The most basic way to do this is writing text to the console. In the previous lesson, we output text to the console using the following program:

If you compile the above program and run it, you will get this output:

The function  is an input function that waits for the user to tap any key on their keyboard and returns a   representing that key. You will learn about input later on in this course.

The line in the above example code that says  is the one that outputs the Hello world! text to the console. is reading what is called a string. In this case, the string is the  part, including the double quotes. You can write almost any text you want.

Escape sequences
tells the console to start a new line. falls under a category called escape sequences. These are commands denoted by the backslash. If you removed "\n" from the first  call in the previous example, the second   would not be printing on a new line, and you would see the following: A65

This is because there is nothing telling the console to put a new line between "A" and "65".

Here is a table of most of the escape sequences.

* For both Hexadecimal and Octal number escape sequences, \nnn or \xnnn evaluates to a character with character code nnn. For example, \101 evaluates to the character A, as A has the character code of 101 in octal format.

Examples

 * Example 1: Some Odyssey

Assignments

 * Exercise 1: Quoth the Raven.

title: C Programming/Basic Output


text:
File:Nightlight.png

Summary
I made this image as an example for DAZ Studio lighting.

title: File:Nightlight.png


text:
User:NTLDR~enwikiversity

Helloooo Wikiversity!
I will be beginning classes for a two-year program in Nursing shortly. The clinical classes will not begin until sometime next year, so my knowledge on the subject is limited. However, I would like to learn from and contribute to School:Nursing as much as I can, so if you have any ideas or tips on expanding the school, please let me know! --NTLDR 01:32, 8 August 2007 (UTC)

title: User:NTLDR~enwikiversity


text:
User talk:NTLDR~enwikiversity

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called NTLDR. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name NTLDR~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:39, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:16, 19 April 2015 (UTC)

title: User talk:NTLDR~enwikiversity


text:
User talk:Pajumed

Hello Pajumed, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 02:15, 8 August 2007 (UTC)

title: User talk:Pajumed


text:
File:Lightanimation.gif

Summary
I made this GIF animation as an example for DAZ Studio lighting.

title: File:Lightanimation.gif


text:
Urantia Book/God

By and large, the Urantia Book is undoubtedly about God. Throughout the book, the authors elucidate on the nature and necessity of God, the relationship of religion to science, and the purpose of existence and evil.

title: Urantia Book/God


text:
User:Dick Tibbetts

I'm a teacher with about 30 years experience in TEFL. My goals as regards the Wikiversity TESOL project are to try and give a clear explanation of the English language, with depth but without confusion, to give examples of different methods and approaches and to see how these can be used in practical teaching.

title: User:Dick Tibbetts


text:
User talk:Dick Tibbetts

Hello Dick Tibbetts, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 04:44, 8 August 2007 (UTC)

title: User talk:Dick Tibbetts


text:
User talk:Weixin

With passion to explore mysterious nature, here should umknown knowledge for me. Wait, here comes me!

title: User talk:Weixin


text:
History of Nepal

Nepal is a country in Asia that has an extensive history.

Discussion questions and essay ideas

 * What are some notable events in the history of Nepal?
 * What historical lessons can be learned from studying the history of Nepal?

Wikipedia

 * History of Nepal
 * Ancient history of Nepal
 * Unification of Nepal
 * Kingdom of Nepal
 * Nepalese democracy movement
 * 1960 Nepal coup d'état
 * Nepalese Civil War
 * History of Asia
 * King of Nepal
 * Politics of Nepal
 * Outline of Nepal
 * Nepal
 * Nepal topics

External

 * Interpreting the India-Nepal border dispute
 * A Reading Guide to Nepalese History

Videos

 * The History of Nepal (grassroots educational video)

title: History of Nepal


text:
Nonprofit management/Incorporate with Secretary of State

When this lesson is done you should be able to answer these questions and be ready to register your organization with the Secretary of State.
 * 1) What is the name and IRS classification of your nonprofit?
 * 2) What state are you incorporating in?
 * 3) Does your nonprofit qualify for a state tax exemption? If yes, what is the form number you will use? How many pages long is the application?
 * 4) Do you need to obtain a state tax identification number? If yes, what is the form number and how many pages long is it?
 * 5) Are you required to register as a charity in your state? What form will you use? How many pages long is the form? What documentation do you need to include?
 * 6) Does your state give real or leased property tax exemptions to nonprofits? If yes, which department do you apply to?

Wikimedia

 * Taxation in the United States


text:
Bloom Clock/Using Bcp3

Using bcp3

From the text list page (using the links created by bcp) :

 * 1) Follow link to commons, pick best available image (should show details of the flowers). If there is no image available, either:
 * 2) Upload an image
 * 3) leave the image and description variables blank
 * 4) Follow the link to "log page", and add
 * (Reminder: if there is no image available, just add the scientific name and the common name. The template will automatically add a "no image available" signal and categorize the page as lacking an image).
 * Note: do not include the "Image:" prefix
 * 1) Save, then follow the instructions for log page creation on the top of the page.
 * 2) If archiving signatures from a text-based clock page, add these to the recent (and/or archived) logs section.
 * 3) If there are no signatures to archive and you wish to log the plant, just add *~ under "recent logs"
 * 4) Return to the plant page, and open edit window
 * 5) Remove all text lines above the section break line
 * 6) Change bcp-u to bcp-s (for a species), bcp-g (for a genus), or bcp-v (for a cultivar or variety)
 * 7) Set the general categories
 * 8) Add regional categories, as appropriate
 * 9) Add month templates to applicable regions
 * 10) Add identifying characteristics (add whatever you know)
 * 11) Press save
 * 12) Return to the text log page, and edit the section
 * 13) If there are signatures to be archive, make sure to add them to the log
 * 14) change bcp to bcp+

title: Bloom Clock/Using Bcp3


text:
File:Matte painting going away v1a.png

Used with the WikiU Film School Lesson on how camera lenses.

title: File:Matte painting going away v1a.png


text:
BCP/Silene vulgaris

Silene vulgaris

title: BCP/Silene vulgaris


text:
BCP/Silene vulgaris/Logs

This page contains recent bloom logs for the plant Silene vulgaris (Bladder Campion).

Recent Logs

 * --Jomegat 15:58, 29 July 2007 (UTC)
 * --Jomegat 02:42, 26 August 2007 (UTC)
 * --Jomegat 23:07, 21 September 2007 (UTC)
 * --Turnvater Jahn 16:16, 20 May 2008 (UTC)
 * Jomegat 02:07, 20 June 2010 (UTC)

=====Archived Logs===== (If there are more than 5 signatures under the recent logs heading, move the oldest logs to under this heading (generally leave the 3 most recent).
 * (no archived logs yet)

title: BCP/Silene vulgaris/Logs


text:
File:Matte painting going away v2a.png

Used with the WikiU Film School Lesson on how camera lenses.

title: File:Matte painting going away v2a.png


text:
BCP/Sagina subulata

Sagina subulata

title: BCP/Sagina subulata


text:
BCP/Sagina subulata/Logs

This page contains recent bloom logs for the plant Sagina subulata (Scottish Moss).

Recent Logs

 * SBJ 11:47, 16 May 2008 (UTC)
 * SBJ11:39, 14 July 2007 (UTC)

=====Archived Logs===== (If there are more than 5 signatures under the recent logs heading, move the oldest logs to under this heading (generally leave the 3 most recent).
 * (no archived logs yet)

title: BCP/Sagina subulata/Logs


text:
BCP/Tradescantia virginiana/Logs

2007 Logs

 * --SB_Johnny | talk 08:36, 4 July 2007 (UTC)

2008 Logs

 * SBJ 08:11, 19 June 2008 (UTC)
 * --SB_Johnny talk 09:59, 10 August 2008 (UTC)

2009 Logs

 * Jomegat 00:43, 12 May 2009 (UTC)
 * Jomegat 02:03, 4 June 2009 (UTC)

Recent Logs

 * Jomegat 23:22, 5 May 2010 (UTC)

title: BCP/Tradescantia virginiana/Logs


text:
BCP/Ambrosia trifida/Logs

2008 Logs

 * (no archived logs yet)

Recent Logs

 * --SB_Johnny | talk 12:23, 28 August 2006 (UTC)
 * --SB_Johnny | PA! 16:28, 4 August 2007 (UTC)
 * --SB_Johnny | PA! 11:38, 8 August 2007 (UTC)
 * --SB_Johnny | talk 09:17, 1 September 2007 (UTC)
 * --SB_Johnny talk 11:22, 10 August 2008 (UTC)
 * --SB_Johnny talk 10:30, 17 August 2008 (UTC)

title: BCP/Ambrosia trifida/Logs


text:
Category:Bloom Clock/Southwestern Oregon Categories

These are the categories for regional data from Southeastern Pennsylvania. These categories should be added to the Bloom Clock Plant pages for both month and flora type.

 * BCP/SWOR/GP: For garden (cultivated) plants
 * BCP/SWOR/IP: For exotic invasive plants (referenced using The Plants of Pennsylvania)
 * BCP/SWOR/NP: For plants native to Pennsylvania (referenced using The Plants of Pennsylvania)


title: Category:BCP/SWOR/8


text:
BCP/Centaurea solstitialis

Centaurea solstitialis

title: BCP/Centaurea solstitialis


text:
BCP/Eschscholzia californica

Eschscholzia californica

title: BCP/Eschscholzia californica


text:
BCP/Eschscholzia californica/Logs

This page contains recent bloom logs for the plant Eschscholzia californica (California Poppy).

Recent Logs

 * Hazel 17:56, 5 August 2007 (UTC)
 * Ngravagna 16:11, 22 May 2008 (UTC)
 * Ngravagna 17:25, 21 June 2008 (UTC)
 * Bastique 00:46, 17 August 2008 (UTC)
 * Bastique 22:01, 26 August 2008 (UTC)

=====Archived Logs===== (If there are more than 5 signatures under the recent logs heading, move the oldest logs to under this heading (generally leave the 3 most recent).
 * (no archived logs yet)

title: BCP/Eschscholzia californica/Logs


text:
BCP/Echinacea purpurea/Logs

This page contains recent bloom logs for the plant Echinacea purpurea (Purple Coneflower).

Recent Logs

 * SBJ 18:24, 4 September 2007 (UTC)
 * --Luai lashire 21:52, 5 September 2007 (UTC)
 * SB Johnny-LM 09:20, 8 September 2007 (UTC)
 * SB Johnny-LM 10:17, 16 October 2007 (UTC)
 * SB Johnny-LM 18:31, 17 November 2007 (UTC)
 * --SB_Johnny | talk 09:54, 14 July 2008 (UTC)
 * Ngravagna 22:47, 2 September 2008 (UTC)
 * --SB_Johnny talk 09:35, 23 September 2008 (UTC)
 * --Clarinetguy097 (discuss • contribs) 19:01, 18 July 2021 (UTC)

=====Archived Logs===== (If there are more than 5 signatures under the recent logs heading, move the oldest logs to under this heading (generally leave the 3 most recent).
 * SBJ 09:34, 27 June 2007 (UTC)
 * --SB_Johnny | talk 10:28, 1 July 2007 (UTC)
 * SB Johnny-LM 10:46, 25 July 2007 (UTC)
 * --SB_Johnny | PA! 13:27, 8 August 2007 (UTC)
 * SB Johnny-LM 10:48, 15 August 2007 (UTC)
 * --Luai lashire 17:57, 15 August 2007 (UTC)
 * SBJ 08:20, 25 August 2007 (UTC)

title: BCP/Echinacea purpurea/Logs


text:
Bloom Clock/Keys/Southwest Oregon/August/All

The plants pictured below have been recorded as blooming during the month of August in Southwest Oregon.


text: Category:Bloom Clock Keys/SWOR

title: Category:Bloom Clock Keys/SWOR


text:
Bloom Clock/Keys/Southwest Oregon/August/All/DPL

This list is used to create a visual key.

 category=BCP/SWOR/8 order=descending ordermethod=lastedit


text:
User talk:Yqbd~enwikiversity

Hello Yqbd, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 14:41, 8 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Yqbd. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Yqbd~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:48, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Yqbd~enwikiversity


text:
User:Mparajuli3

history of nepal

title: User:Mparajuli3


text:
User talk:Mparajuli3

Hello Mparajuli3, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 14:51, 8 August 2007 (UTC)

title: User talk:Mparajuli3


text: File:Wakeup.ogg

title: File:Wakeup.ogg


text:
User:Designgrl

'''I'm Lisa Beeson, aka "designgrl". I've been designing curriculum for classroom and the web for just about ten years now. I also have managed the implementation and day-to-day operations of several LMSs which means I'm very "comfy" with them.'''

I adore learning design and consider myself a lifelong student.

Along with Wikiversity I am also a learning developer for OpenOffice.org and a member of the Moodle, eLearning Guild, and PeLDA developer communities.

Glad to be here!

title: User:Designgrl


text:
User talk:Designgrl

Hello Designgrl, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 15:54, 8 August 2007 (UTC)

Image copyright problems
Your currently affected uploads are listed below. For each, please provide a source and license template. Thanks. – Mike.lifeguard | @en.wb 23:51, 13 February 2008 (UTC)
 * 2007-08-08 15:45, Designgrl: SimpsonizedLisa.jpg: untagged! ; 0 (0)
 * 2007-08-08 15:52, Designgrl: Lisa's Birthday Fiesta March 2007.jpg: untagged! ; 2 (0)

title: User talk:Designgrl


text:
Pakistan/A brief history

Welcome to this learning project about Pakistan- a brief history

This is a suggested list of headings to organise a learning project - please see Directions for use for more information. Please delete any headings (etc.) if you feel they are not relevant, and rearrange as you see fit.

Learning Project Summary

 * Time investment:Work in progress, unsure at this point
 * Assessment suggestions:essays (etc.) shall be set in due course
 * Portal:(Social Sciences)
 * School:(School of History)
 * Department: Indic History
 * Level: Basic

Content summary
This learning project is all about the country that has recently been at the forefront of the press.

Goals
This learning project aims to.. -- Above is a basic outline- more topics shall be covered but these are the main points.
 * Educate about the basic origins of Pakistan
 * Pre-partition
 * Mughals (see Mughal India)
 * Founding Fathers
 * Muhammad Ali Jinnah (Quaid-i-Azam)
 * Discuss modern Pakistan (on a daily basis)
 * Inform about poltical and socio-economic issues.
 * Relations with the West
 * The War on Terror

Learning materials
(Including some internal materials- for mroe see the box below)

 * History of Pakistan (Wikipedia)
 * History of the Pakistani military-Wikipedia
 * Pakistan- Wikipedia
 * Gives a list of all Pakistna related topics on Wikipedia
 * About Pakistan (Wikibooks)
 * CIA factfile on Pakistan (Wikibooks)
 * BBC country profile
 * Pakistani Government Website

(more shall be added in due course, if more sources are needed the external links sections of the Wikipedia articles are good.)

Lessons
Introduction to Pakistan

Active participants
Active participants in this Learning Group
 * geminibubblegum

{| width="100%" style="background:#FFFCCF; border:1px solid #DAA520"

Learning materials
Learning materials and learning projects are located in the main Wikiversity namespace. Simply make a link to the name of the lesson (lessons are independent pages in the main namespace) and start writing!

You should also read about the Wikiversity:Learning model. Lessons should center on learning activities for Wikiversity participants. Learning materials and learning projects can be used by multiple projects - and you are encouraged to cooperate with other departments that use the same learning resource.

 * }

title: Pakistan/A brief history


text:
User talk:Geminibubblegum

Hello Geminibubblegum, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 18:44, 8 August 2007 (UTC)

title: User talk:Geminibubblegum


text:
Talk:Astrochemistry

Message To Founder of Department of Astrochemistry

 * 1) In ¶ 01 Edits have been made Linking the Topic "Astrochemistry" to the School of Chemistry and to the Wikipedia article "Universe".
 * 2) In ¶ 02 Edits have been made again Linking to the Wikipedia article "Universe".
 * 3) In ¶ 02 The Question arises: What is the meaning of "materia"?  Could it be material or is it something other, such as even materia?
 * 4) In ¶ 02 The Question arises: What is a ""chemistry object""?  Could the sentence mean somethng like "Our Sun is made of chemicals.  The meaning is not clear.
 * 5) Re ¶ 03: Sara Goudarzi's February, 2007 article "Milky Way Fortified with Calcium" appears to be Copyrighted Material found at http://www.space.com/scienceastronomy/070212_universe_calcium.html .  If so, the Text probably could not be properly be used at Wikiversity.  The use of Copyrighted Material material on Wikiversity Pages could result inan Issue Coming Up Regarding the use of Copyrighted Material.  Accordingly, the Article has been Removed and Replaced with a Link.  In all your Work, please Cite the Source and please comply with all Wikiversity Policies.

(s) Dionysios  (talk), Date: 2007-08-08 (August 8, 2007) Time:  2011 UTC

title: Talk:Astrochemistry


text:
User:Nostalgia~enwikiversity

Uh... Hi? I am typing this, and I am still typing this, yet, still typing this, and guess whos typing this? Thats right, me!

title: User:Nostalgia~enwikiversity


text:
File:SBTDS Storyboard Shot 01.png

Used with the Wikiversity Film School Thumbnail Storyboard Pop Quiz.

title: File:SBTDS Storyboard Shot 01.png


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title: File:SBTDS Storyboard Shot 08.png


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File:SBTDS Storyboard Shot 21.png

Answer for Pop Quiz at WikiU Film School Lesson on storyboarding.

title: File:SBTDS Storyboard Shot 21.png


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File:SBTDS Storyboard Shot 09.png

Used with the Wikiversity Film School Thumbnail Storyboard Pop Quiz.

title: File:SBTDS Storyboard Shot 09.png


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File:SBTDS Storyboard Shot 19.png

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File:Briana.ogg

Summary
Original source: http://wikimania.tw/streaming/

http://wikimania.tw/streaming/room328_day_3.ogg

title: File:Briana.ogg


text: File:RavenConsole.jpg

title: File:RavenConsole.jpg


text:
C Programming/Basic Output/Exercise 1

This exercise is fairly simple and the purpose is to help familiarize yourself with outputting text to the console, which you will be doing a lot of, and helping you to also familiarize yourself with the escape sequences.

You're assignment is to write the C source code that will output this:

Use  to write the text to the console. Use one call of  for each line. This means you'll have to put a newline command at the end of each line. You could cram this all into one call of  but that would look very cluttered and unorganized.

Don't forget that single and double quotes need to be escaped or else you will get compilation errors.

For the last line, use 3 horizontal tab escape sequences before the text.

title: C Programming/Basic Output/Exercise 1


text:
User talk:Remi/May '07 – July '07

username change
"change the custodian status from this username User:Remi0o to User:Remi?" <-- I suggest that you ask User:Sebmol. --JWSchmidt 22:53, 7 May 2007 (UTC)

 * Hmm...I think it might be better to have sebmol rename this "remi" account to something...say, "remi2", and then have sebmol do an usurpation of some sort to move your "Remi0o" account to "Remi". I think it will only be successful provided that you don't edit from "Remi", so for now just log on to your old Remi0o account until everything is as it should be. --HappyCamper 23:19, 7 May 2007 (UTC)

re: bloom clock
Probably would help... depending on if the plant can be identified :). I'm in a bit of a bind because while I finally more or less figured out how to structure the clock (through trying to use it), I really have no time to do the cleanups at this time of year.

To make the data useful, just sign up on Bloom clock project/Contributors, so a bot can eventually use your location to organize the data. (Latitude isn't enough... for example, New York City is actually south of Barcelona, Spain, but the flowering times in Southern Spain will be far different than those in New York). --SB_Johnny | talk 11:53, 9 May 2007 (UTC)

Grants update
I've started work on a serious proposal here. It's a simple project, but hopefully it's got enough nice stuff in it to attract attention and garner support. I'd appreciate feedback and whatnot ... Thanks. Historybuff 19:11, 9 May 2007 (UTC)

Username stuff...
Heh, I recently changed accounts on Wikibooks, but due to transwiki it was more or less impossible to actually just "move" the account, so I just locked up the old one.

If you haven't used "Remi" yet on Wikibooks, I'd be happy to change your username there for you. --SB_Johnny | talk 10:50, 10 May 2007 (UTC)
 * Already did with redirects... but thank you. --Remi 05:19, 12 May 2007 (UTC)

Catagories
Hey Remi, I see you've been categorizing some wikipages and I had a suggestion about it. You recently tagged one of my pages (Russian Revolution/Week 7) with the Soviet Studies category. I would suggest you categorize the entire class under Soviet Studies since there are six other weeks on the same subject and it would be more efficient to just tag the ‘homepage’ for the class. Therefore when you are looking at the list later on, you don’t have 10+ Russian Revolution pages. Just a though…
 * Thank you for the feedback. It seems like that would make things more navigable. --Remi 05:18, 12 May 2007 (UTC)

Pro template spacing
I might be the guilty party. I categorised a lot of templates recently. McCormack 07:17, 12 May 2007 (UTC)
 * Oh no, my mistake; I just put it all on one line and it worked. I was going to remove my comment... and fix Template:Pro. Thank you for categorizing pages by the way; I think in the long run it will really be advantageous for the sustainability of the project. --Remi 07:22, 12 May 2007 (UTC)

Welcome
Hi Remi, I noticed you added your own version of the welcome template to User talk:ShaenaPete. I like it, and I'm wondering if you'd like to work on a new template - many people (myself included) have long thought that Template:Welcome needs to be adapted. What do you think? Do you have a templatised version of that message you added? Cormaggio talk 15:45, 12 May 2007 (UTC)
 * I adapted that from Wikibooks... There should be a templatized version around here. I think you are right on that mark, though. The current template (which I went back to using for consistency) should maybe integrate some features of the Wikibooks template.

 * If my memory serves me, what I like about the Wikibooks welcome template is that it basically shows users what what they need to know to get involved in a rather functional, logical, and sequential order (I tweaked it a bit at some point). While the current welcome being used here is good, it could maybe use some tweaking to better convey to new users a digestible amount of information that will be useful. --Remi 04:53, 13 May 2007 (UTC)

 * There's a sandbox for you to get started on at Template:Welcome/new or, if you prefer, you can make suggestions on either that page's or the existing template's talk pages. Cormaggio talk 21:43, 13 May 2007 (UTC)
 * Hiya, Template:Welcome/Unstable seems redundant with Template:Welcome/new.. Cormaggio talk 11:30, 16 May 2007 (UTC)
 * You're right. Thank you for the heads up. --Remi 05:07, 9 June 2007 (UTC)

Username and custodian
Hi Remi, I've made this account (User:Remi) a custodian, but I can't change the status of User:Remi0o, as bureaucrats do not have that privilege. This needs to be done by a steward instead - you can request this at Requests for permissions. Cheers, Cormaggio talk 20:49, 13 May 2007 (UTC)
 * Thank you. I will go ahead and give the stewards notice of the change. --Remi 21:48, 13 May 2007 (UTC)
 * No problem. Which account are you going to edit from from now on? Are you going to use both? Cormaggio talk 21:50, 13 May 2007 (UTC)
 * Most likely just this one. I do not forsee any reason to switch back. --Remi 21:52, 13 May 2007 (UTC)
 * That makes most sense. I only asked because you edited from the other one just now. ;-) Cormaggio talk 22:00, 13 May 2007 (UTC)

Topic:Introduction to Psychology
I formated Topic:Introduction to Psychology as a content development project. --JWSchmidt 00:11, 17 May 2007 (UTC)

custodianship
You are now a full custodian. Thanks for serving the Wikiversity community as a custodian. --JWSchmidt 14:49, 20 May 2007 (UTC)
 * Thank you for giving me the opportunity to help out. Wikiversity is an interesting project to say the least! : ) --Remi 05:09, 9 June 2007 (UTC)

Thanks for working on the categories
From Categories/2007/Week21: This week our barnstar goes to Remi for these achievements

I've started a project at Categories for helping you and others with cleaning up the Category Tree. There's lots of work to do, as you obviously know, but I thought this might provide a structured approach, while showing others how it's done on a continuous weekly basis. Since you know a bit about the process, your input would be quite appreciated. -- CQ 01:30, 22 May 2007 (UTC)
 * Thank you for starting that page. I find categorizing pages oddly satisfying. : ) I will try to give some input... --Remi 05:37, 22 May 2007 (UTC)
 * Hey Remi! Just to chime in to say you're awesome! :-) --HappyCamper 01:00, 23 May 2007 (UTC)
 * Thank you. You're not so bad yourself. : ) --Remi 05:10, 9 June 2007 (UTC)

Wikiversitarians
Hi Remi. I believe you were asking somewhere about the correct word for someone who edits at Wikiversity - i.e. an equivalent for Wikipedian. I've done a little research, and I believe one can say, definitively, that there is only one etymologically correct word.

 * Original word from which Wikiversity is derived: university.
 * The obscure but correct and well documented adjective from university: universitarian (no other forms known).
 * A person who is a member of a university, if not referred to as student/staff, could therefore be correctly referred to by turning the adjective into a noun: a universitarian.
 * By analogy, the etymologically correct word you are looking for is: wikiversitarian.

 * -- McCormack 06:52, 28 May 2007 (UTC)

 * I thought it was "wikiversitan", actually :)--SB_Johnny | talk 12:15, 28 May 2007 (UTC)

 * Take a look at Developing Wikiversity through action research/Wikiversity identity. :-) Also, if you get a chance, could you take a look at Template:Welcome/new and see what you think? --HappyCamper 17:08, 2 June 2007 (UTC)
 * What to call people who use Wikiversity? I'm staking out of that. : ) As for the new welcome page, I think it looks really good and will make it easier for new users to get the information they need to get involved. --Remi 18:27, 2 June 2007 (UTC)

converting content development projects to lessons
Why convert content development projects in the "Topic:" namespace into main namespace pages? In many cases, doing so not only destroys a content development project but it also disrupts the function and organization of one or more "School:" and/or "Topic:" pages that link to the destroyed content development page. If you find actual learning resources on a page in the "Topic:" namespace, the solution is to move that content to a page in the main namespace and leave the content development page in existence for further development of content. It is true that some pages in the "Topic:" namespace have never been formatted as content development pages, but that does not mean that those content development pages should be destroyed. Those pages should just be correctly formated as content development projects. --JWSchmidt 20:27, 15 June 2007 (UTC)

title: User talk:Remi/May '07 – July '07


text:
User talk:Nostalgia~enwikiversity

Hello Nostalgia, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --Remi 00:10, 9 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Nostalgia. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Nostalgia~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:45, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:Nostalgia~enwikiversity


text:
User talk:Zorak~enwikiversity

Hello Zorak, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 03:16, 9 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Zorak. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Zorak~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:48, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Zorak~enwikiversity


text:
Openmoko/Intro to the Neo

I have a Neo1973 and I've started to play with it.

Still early on, but I've managed to flash it (from a Mac) and get it to boot. I've SSH'ed into it, and I've played with various bits. Still a lot to learn, but it's very exciting and should be fun once it gets going.

I've now managed to make a GSM call and register to the network in my area. I tried to get Bluetooth working, but the instructions imply you need to be USB connected (not sure if the instructions are just USB centric).

I've got sound output to work, but only have some hints as to how sound input should work. It's still too early to "eat my own dogfood" on the device, but there is lots to do, and it's all exciting.

Aug 18 - Finally got a matching kernel and rootfs flashed, and have managed to bring up (but not get working) bluetooth. I've also got a basic build enviornment installed, now I have to build up a full version and see if I can get an image ready to be installed.

title: Openmoko/Intro to the Neo


text:
Malaria

Malaria is a vector-borne infectious disease that is widespread in tropical and subtropical regions, including parts of the Americas, Asia, and Africa. Each year, it causes disease in approximately 650 million people and kills between one and three million, most of them young children in Sub-Saharan Africa. Malaria is commonly-associated with poverty, but is also a cause of poverty and a major hindrance to economic development.

Malaria is one of the most common infectious diseases and an enormous public-health problem. The disease is caused by protozoan parasites of the genus Plasmodium. The most serious forms of the disease are caused by Plasmodium falciparum and Plasmodium vivax, but other related species (Plasmodium ovale, Plasmodium malariae, and sometimes Plasmodium knowlesi) can also infect humans. This group of human-pathogenic Plasmodium species is usually referred to as malaria parasites.

Malaria parasites are transmitted by female Anopheles mosquitoes. The parasites multiply within red blood cells, causing symptoms that include symptoms of anemia (light headedness, shortness of breath, tachycardia etc.), as well as other general symptoms such as fever, chills, nausea, flu-like illness, and in severe cases, coma and death. Malaria transmission can be reduced by preventing mosquito bites with mosquito nets and insect repellents, or by mosquito control by spraying insecticides inside houses and draining standing water where mosquitoes lay their eggs.

No vaccine is currently available for malaria; preventative drugs must be taken continuously to reduce the risk of infection. These prophylactic drug treatments are often too expensive for most people living in endemic areas. Most adults from endemic areas have a degree of long-term recurrent infection and also of partial resistance; the resistance reduces with time and such adults may become susceptible to severe malaria if they have spent a significant amount of time in non-endemic areas. They are strongly recommended to take full precautions if they return to an endemic area. Malaria infections are treated through the use of antimalarial drugs, such as quinine or artemisinin derivatives, although drug resistance is increasingly common.

Malaria parasites
Malaria is caused by protozoan parasites of the genus Plasmodium (phylum Apicomplexa). In humans malaria is caused by P. falciparum, P. malariae, P. ovale, and P. vivax. However, P. falciparum is the most important cause of disease and responsible for about 80% of infections and 90% of deaths. Parasitic Plasmodium species also infect birds, reptiles, monkeys, chimpanzees and rodents. There have been documented human infections with several simian species of malaria, namely P. knowlesi, P. inui, P. cynomolgi, P. simiovale, P. brazilianum, P. schwetzi and P. simium; however these are mostly of limited public health importance. Although avian malaria can kill chickens and turkeys, this disease does not cause serious economic losses to poultry farmers. However, since being accidentally introduced by humans it has decimated the endemic birds of Hawaii, which evolved in its absence and lack any resistance to it.

Pathogenesis
Malaria in humans develops via two phases: an exoerythrocytic (hepatic) and an erythrocytic phase. When an infected mosquito pierces a person's skin to take a blood meal, sporozoites in the mosquito's saliva enter the bloodstream and migrate to the liver. Within 30 minutes of being introduced into the human host, they infect hepatocytes, multiplying asexually and asymptomatically for a period of 6–15 days. During this so-called dormant time in the liver, the sporozoites are often referred to as hypnozoites. Once in the liver, these organisms differentiate to yield thousands of merozoites which, following rupture of their host cells, escape into the blood and infect red blood cells, thus beginning the erythrocytic stage of the life cycle. The parasite escapes from the liver undetected by wrapping itself in the cell membrane of the infected host liver cell. Within the red blood cells the parasites multiply further, again asexually, periodically breaking out of their hosts to invade fresh red blood cells. Several such amplification cycles occur. Thus, classical descriptions of waves of fever arise from simultaneous waves of merozoites escaping and infecting red blood cells.

Some P. vivax and P. ovale sporozoites do not immediately develop into exoerythrocytic-phase merozoites, but instead produce hypnozoites that remain dormant for periods ranging from several months (6–12 months is typical) to as long as three years. After a period of dormancy, they reactivate and produce merozoites. Hypnozoites are responsible for long incubation and late relapses in these two species of malaria. This is conventional view. However, it is now becoming recognized that relapse-like recurrent P. vivax malaria might frequently have a hypnozoite-independent origin(s). The parasite is relatively protected from attack by the body's immune system because for most of its human life cycle it resides within the liver and blood cells and is relatively invisible to immune surveillance. However, circulating infected blood cells are destroyed in the spleen. To avoid this fate, the P. falciparum parasite displays adhesive proteins on the surface of the infected blood cells, causing the blood cells to stick to the walls of small blood vessels, thereby sequestering the parasite from passage through the general circulation and the spleen. This "stickiness" is the main factor giving rise to hemorrhagic complications of malaria. High endothelial venules (the smallest branches of the circulatory system) can be blocked by the attachment of masses of these infected red blood cells. The blockage of these vessels causes symptoms such as in placental and cerebral malaria. In cerebral malaria the sequestrated red blood cells can breach the blood brain barrier possibly leading to coma.

Although the red blood cell surface adhesive proteins (called PfEMP1, for Plasmodium falciparum erythrocyte membrane protein 1) are exposed to the immune system they do not serve as good immune targets because of their extreme diversity; there are at least 60 variations of the protein within a single parasite and perhaps limitless versions within parasite populations. Like a thief changing disguises or a spy with multiple passports, the parasite switches between a broad repertoire of PfEMP1 surface proteins, thus staying one step ahead of the pursuing immune system.

Some merozoites turn into male and female gametocytes. If a mosquito pierces the skin of an infected person, it potentially picks up gametocytes within the blood. Fertilization and sexual recombination of the parasite occurs in the mosquito's gut, thereby defining the mosquito as the definitive host of the disease. New sporozoites develop and travel to the mosquito's salivary gland, completing the cycle. Pregnant women are especially attractive to the mosquitoes, and malaria in pregnant women is an important cause of stillbirths, infant mortality and low birth weight.

title: Malaria


text:
Introduction in tropical diseases

Around the Rio Dulce area, the occurrence of tropical diseases is quite rare but isolated cases of intestinal problems, malaria and dengue fever will occasionally crop up. Some years ago there was a minor outbreak of cholera in the Fronteras area. The solution is simply this: 1) Under no circumstances should you drink the river water. 2) Don't bathe in the river near or downstream of heavily populated areas, ie. Fronteras.

Some have expressed a general interest in tropical diseases and have complained of the lack of information about such diseases. To satisfy this need the following general overview of tropical diseases worldwide is included here. Some of the diseases covered below do not occur in Central America but are included to satisfy the curious. Readers with a taste for the macabre may be especially interested in the news-clip about the Ebola virus.

Tropical diseases are illnesses that either occur uniquely in tropical and subtropical regions (which is rare) or, more commonly, are either more widespread in the tropics or more difficult to prevent or control. The tropics are more problematic for certain diseases for two reasons: 1) Tropical climates are more conducive to certain diseases. 2) Areas of poverty and primitive sanitation conditions are more common in the tropics. The agencies most concerned with improving health in tropical countries include the World Health Organization (WHO) and the Edna McConnell Clark Foundation.

Diseases Most Prevalent in the Tropics

The most important diseases in the tropical regions of Southeast Asia, Africa, and South America are malaria, schistosomiasis, leprosy, filariasis, trypanosomiasis, and leishmaniasis. Although effective chemotherapy and insecticides have reduced or eliminated malaria in most of the western hemisphere, these measures have been less successful in Asia. Both the infecting parasite and its mosquito carrier have become resistant to current drugs and 200 million persons are estimated to have malaria in tropical areas. In sub-Saharan Africa some 1 million children under five die of the disease each year.

Schistosomiasis has never been common in temperate climates, but it affects 125 million persons worldwide, of whom approximately 20 percent are at least partly disabled by the disease. Praziquantel, a highly effective new drug, is now available for treatment of schistosomiasis. Leprosy has also always been more common in tropical than in moderate climates, and about 11 million persons in the world have this illness. In endemic areas many severe cases of leprosy are now resistant to the drug first used against it, and newer, more expensive therapy must be employed. Filariasis is a common tropical debilitating illness caused by infection with roundworm larvae. Trypanosomiasis, which results from infection with a protozoan parasite, has caused 10 million cases of human sickness in Africa alone. A related protozoan in South America causes a less deadly form of trypanosomiasis called Chagas' disease. Leishmaniasis is also a result of worm infection, and in its Asian and African forms the disease can damage the internal organs.

Additional Health Problems

Although tuberculosis is largely under control in developed countries, it is still a considerable public health problem in much of the world and is responsible for about half a million deaths annually, 75 percent of them in Asia. Other diseases for which treatment is available but which are still common in developing countries include cholera, yellow fever, yaws, and amoebic dysentery (see Dysentery).

Two forms of cancer, Burkitt's lymphoma and liver cancer, are very common in Africa and Asia, respectively, although rare in temperate zones. Burkitt's lymphoma is thought to be due to a combination of massive infection with a virus early in life and malaria in adulthood. Liver cancer may be caused by infection with the hepatitis B virus.

As many as 25 million persons have become blind from preventable diseases in tropical countries. These diseases include xerophthalmia, due to lack of vitamin A in the diet; onchocerciasis, or river blindness, an infection of the skin by filarial larvae that may also affect the conjunctiva of the eye; and trachoma, a chronic conjunctival infection caused by the parasitic bacterium Chlamydia trachomatis, which is transmitted by flies or through close personal contact.

Finally, a number of severe virus-caused fevers that were identified during the 1970s are found predominantly in tropical regions. These diseases include Lassa, Ebola, Marburg, Bunya, and Chikungunya fevers, some of which cause death by hemorrhage (hemorrhagic fever). One member of this family, dengue virus (see Dengue Fever), was known for many years but has recently spread to the Caribbean and Mexico. All these diseases are rare.

Factors Which Aggravate Tropical Diseases

The severity of diseases in tropical areas is due to widespread poverty and poor sanitation as well as climatic influences. That is, because of low national incomes, most developing countries cannot afford to buy vaccines to prevent poliomyelitis, measles, and yellow fever. Only about 10 percent of the 80 million children in poor countries have been immunized against diphtheria, whooping cough, and tetanus, and such countries cannot afford to distribute drugs against tuberculosis or leprosy. Poverty is a condition that also leads to malnutrition, which makes people more susceptible to disease.

Poor sanitation is especially to blame for the spread of cholera, in which the infecting agent is transmitted through contaminated sewage; and schistosomiasis, in which the intermediate vector, a snail, lives in contaminated water.

Climate indirectly makes disease in tropical regions more severe by reducing agricultural production, which increases the risk of malnutrition. In a more direct way, hot weather and humid forests favor growth of the flies and mosquitoes that transmit malaria, yellow fever, dengue fever, trachoma, trypanosomiasis, and onchocerciasis.

 * George-Cristian Potrivitu

title: Introduction in tropical diseases


text:
Novel economic mechanisms

This project is intended to explore and possibly implement novel economic mechanisms that are potentially beneficial to all members of society.

Discussion questions

 * How might microinvestments be approached in order to allow large sums of capital to be accumulated and then have collaboratively decided business decisions be executed?

title: Novel economic mechanisms


text:
User:Jana~enwikiversity

Background
I was born in Kladno and I still live there. My native language is Czech. I can speak: I would love to learn Spanish. I studied at the Institute of Chemical Technology in Prague - specialization brewing.
 * upper intermediate English
 * intermediate German
 * beginner Russian

Positions and Reponsibilities
I work for Client Servers Solutions Group as a technical writer. My responsibilities are:
 * 1) interviewing programmers
 * 2) updating and creating documents
 * 3) co-operation with the  technical writers group

Goals
My professional goal is to become a certificate technical writer and start writing more document types.

About my personal goals: buy a house, become an instructor of fitness and nutrition consultant and I would like to learn programming. If it would be possible I'll start with box.

Enviroment
I'm surrounded by many nations in our company and that's great. But I don't like these open space offices, because there is lot of noise. Sometimes it is almost impossible to work. I would like to work in a smaller office only with my colleagues.

Interests
I love doing a lot of things, but the following are the most important for me:
 * softball
 * spinning
 * bodybuilding
 * reading
 * animals (I love my three cats and all other pets that I have)

title: User:Jana~enwikiversity