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User:Egm6321.f10.team4.Yoon/HW1

= HW Problem 1 - First and Second Order Derivative Derivations =

Given
 The nominal motion of the Maglev train can be modeled by the following equation

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$$\displaystyle f(Y^1 (t),t) $$
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Find
Derive the given modeling equations through first and second order derivatives.

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Axial deformation under moving wheel/guideway $$ \frac{d}{dt} f \left(Y^1(t),t\right) = \frac{\partial f}{\partial s} \left(Y^1(t),t\right) \dot Y^1(t) + \frac{\partial f}{\partial t} \left(Y^1(t),t\right) $$

 * (Eq.1)

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Transverse deformation of wheel/guideway $$\frac{d^2}{dt^2} f\left(Y^1(t),t\right) = f_{,s} \left(Y^1(t),t\right) \ddot Y^1 + f_{,ss} \left(Y^1(t),t\right) (\dot Y^1)^2 + 2 f_{,st} \left(Y^1(t),t\right) \dot Y^1 + f_{,tt} \left(Y^1(t),t\right) $$

 * (Eq.2)

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Solution for (Eq.1)
First, equating the function to a dummy variable $$s$$ and taking the time derivative

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$$\displaystyle Y^1(t)=s $$
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\frac{d}{dt} f \left(Y^1(t),t\right) = \frac{d}{dt} f \left(s,t\right) $$

By the chain rule, the equation becomes
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$$\displaystyle \frac{d}{dt} f \left(s,t\right) = \frac{\partial f}{\partial s} \underbrace {\frac{\partial s}{\partial t}}_{=\dot Y^1(t)}+ \frac{\partial f}{\partial t} \underbrace{\frac{\partial t}{\partial t}}_{=1} $$
 * (Eq.3)

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(Equation 3) can be rewritten by reverting function $$f$$ back to its original form:

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$$ \frac{d}{dt} f \left(Y^1(t),t\right) = \frac{\partial f}{\partial s} \left(Y^1(t),t\right) \dot Y^1(t) + \frac{\partial f}{\partial t} \left(Y^1(t),t\right) $$ (Eq.4)
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Solution for (Eq.2)
Substituting the left side of (Eq.2) to the solution of (Eq.1) yields

\frac{d^2}{dt^2} f\left(Y^1(t),t\right) = \frac{d}{dt}\left( \frac{d}{dt} f\left(Y^1(t),t\right)\right) $$

Applying the chain rule again ahead of (Eq.2) and the dummy s

= \frac{\partial}{\partial s}\left( \frac{d}{dt} f\left(Y^1(t),t\right)\right) + \frac{\partial}{\partial t}\left( \frac{d}{dt} f\left(Y^1(t),t\right)\right) = \frac{\partial}{\partial s}\left( \frac{\partial f}{\partial s} \frac{\partial s}{\partial t}+ \frac{\partial f}{\partial t} \right) + \frac{\partial}{\partial t}\left( \frac{\partial f}{\partial s} \frac{\partial s}{\partial t} +  \frac{\partial f}{\partial t} \right) $$

=\frac{\partial^2 f}{\partial s^2} \left(\frac{\partial s}{\partial t}\right)^2 + \frac{\partial^2 f}{\partial s \partial t} \frac{\partial s}{\partial t} \underbrace{\frac{\partial t}{\partial t}}_{=1} + \frac{\partial f}{\partial s} \underbrace{\frac{\partial^2 s}{\partial s \partial t}}_{=0}\frac{\partial s}{\partial t} + \frac{\partial f}{\partial s} \frac{\partial^2 s}{\partial t^2}\underbrace{\frac{\partial t}{\partial t}}_{=1} + \frac{\partial^2 f}{\partial s\partial t}\frac{\partial s}{\partial t} + \frac{\partial^2 f}{\partial t^2} \underbrace{\frac{\partial t}{\partial t}}_{=1} $$

Collecting terms in the above equation by reduced order of y fucntion.

=\frac{\partial^2 f}{\partial s^2} \left(\frac{\partial s}{\partial t}\right)^2 + 2\frac{\partial^2 f}{\partial s \partial t} \frac{\partial s}{\partial t} + \frac{\partial f}{\partial s} \frac{\partial^2 s}{\partial t^2} +  \frac{\partial^2 f}{\partial t^2} $$

By transforming the Leivniz expression to Lagrange expression such as X,y instead of dx/dy,

\frac{d^2}{dt^2} f\left(Y^1(t),t\right) =f_{,ss} \underbrace{\left(\frac{\partial s}{\partial t}\right)^2}_{\dot {Y^1}^{2}} + 2f_{,st} \underbrace{\frac{\partial s}{\partial t}}_{\dot Y^1} + f_{,s}\underbrace{\frac{\partial^2 s}{\partial t^2}}_{\ddot Y^1} + f_{,tt} $$ Finally, we can complete the desired solution by displacing the dummy variable s to the original function.
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$$ \frac{d^2}{dt^2} f\left(Y^1(t),t\right) =f_{,ss} \left(\dot Y^1\right)^2 + 2f_{,st} \dot Y^1 + f_{,s}\ddot Y^1 + f_{,tt} $$
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Similaritry to/Difference from Derivation of Coriolis Force
in Rotating coordinate systems(i.e. the earth) To compare second order differential equation to derivation of Coriolis effect, the idea of fictitious forces is used. If the rotation of frame B can be described as a vector Ω pointed along the axis of rotation

 * $$ |\boldsymbol{\Omega} | = \frac {d \theta }{dt} = \omega (t) \, $$

then the time derivative of the three unit vectors consisting the vector Ω, defined as an observed frame with respect to frame A is

 * $$ \frac {d \mathbf{u}_j (t)}{dt} = \boldsymbol{\Omega} \times \mathbf{u}_j (t) \, $$

and

 * $$\frac {d^2 \mathbf{u}_j (t)}{dt^2}= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j+ \boldsymbol{\Omega} \times \underbrace{\left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)}_{\frac {d \mathbf{u}_j (t)}{dt}}, $$

as is verified using the properties of the vector cross product. These derivative formulas now are applied to the relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with time-varying angular velocity ω ( t ). From the previous section, where subscript A refers to the inertial frame and B to the rotating frame, setting aAB = 0 to remove any translational acceleration, and focusing on only rotational properties

 * $$ \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} $$ $$+ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ ,$$
 * $$\mathbf{a}_A=\mathbf{a}_B +\ 2\ \sum_{j=1}^3 v_j \boldsymbol{\Omega} \times \mathbf{u}_j (t)\ $$ $$+\   \sum_{j=1}^3 x_j \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j  \  + \sum_{j=1}^3 x_j \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)\ $$
 * $$=\mathbf{a}_B $$ $$+ 2\ \boldsymbol{\Omega} \times\sum_{j=1}^3 v_j \mathbf{u}_j (t) \  $$ $$+ \frac{d\boldsymbol{\Omega}}{dt} \times \sum_{j=1}^3 x_j \mathbf{u}_j  \   $$ $$+\ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \sum_{j=1}^3 x_j  \mathbf{u}_j (t) \right)\ .$$

Collecting terms,

 * $$\mathbf{a}_A=\mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ $$ $$+ \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}_B  \  + \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ .$$

Technically, the aceleration aA in the inertial frame A is not exact with the acceleration aB seen by observers in the rotational frame B but has several additional geometric acceleration terms associated with the rotation of B. Rearranging the above equation by the acceleration B

\mathbf{a}_{B} = \mathbf{a}_A - 2 \boldsymbol\Omega \times \mathbf{v}_{B} - \boldsymbol\Omega \times (\boldsymbol\Omega \times  \mathbf{x}_B )  - \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_B \. $$

The force upon the object in the B coordinate system with respect to observers in the rotating frame A is FB = m aB. If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional force Ffict is present, so the end result is FB = FA + Ffict. Thus, the fictitious force used by observers in B to get the correct behavior of the object from Newton's laws equals:

\mathbf{F}_{\mathrm{fict}} = \underbrace{- 2 m \boldsymbol\Omega \times \mathbf{v}_{B}}_{Coriolis force} \underbrace{- m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B )}_{centrifugal force} $$ $$ \underbrace{\ - m \frac{d \boldsymbol\Omega }{dt} \times \mathbf{x}_B}_{Euler force=0}  \. $$

When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero.

= HW1.problem 6 =

Given

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$$ \left(1-x^2 \right)\cdot y''-2x\cdot y'+2y=0$$
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Verify

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$$L_{2} \left( {y}^{1}_{H}\right)=L_{2} \left( {y}^{2}_{H}\right)=0$$
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Solution

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To find $$L_{2} \left( {y}^{1}_{H}\right)=0$$
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Differntiating the first given solution, and differentiating again to find

$${y}^{1}_{H}\left(x \right)=x$$

$$\frac{dy}{dx}=1,$$

$$ \frac{d^2y}{dx^2}=0$$

These can then be substituted into the given differential equation (1) to find

$$ L_{2} \left( {y}^{1}_{H}\right)=\left(1-x^2 \right)\cdot \left( 0 \right)-2x\left( 1 \right)+2x=0$$

To find $$L_{2} \left( {y}^{2}_{H}\right)=0$$

Repeatly differentiated the second solution to find

$${y}^{2}_{H}\left(x \right)= \frac{x}{2}log \left( \frac{1+x}{1-x} \right)-1$$

$$\frac{dy}{dx}= \frac{1}{2}log \left( \frac{1+x}{1-x} \right)+ \left( \frac{1}{1+x}+ \frac{1}{1-x}\right)=\frac{1}{2}log \left( \frac{1+x}{1-x} \right)+ \left(  \frac{x}{1-x^2}\right),  $$

$$\frac{d^2y}{dx^2}= \frac{1}{2} \left( \frac{1}{1+x}+ \frac{1}{1-x} \right)+ \frac{1\cdot \left( 1-x^2\right)-x\cdot \left( -2x\right)}{\left( 1-x^2 \right)^2}=\frac{1}{ \left(1-{x}^{2} \right)}+ \frac{1+x^2}{\left( 1-x^2 \right)^2}$$

Again, substituted into the given differential equation (1) to find

$$L_{2} \left( {y}^{2}_{H}\right)=\left(1-x^2 \right)\cdot \left( \frac{1}{ \left(1-{x}^{2} \right)}+ \frac{1+x^2}{\left( 1-x^2 \right)^2} \right)-2x\left( \frac{1}{2}log \left( \frac{1+x}{1-x} \right)+ \left( \frac{x}{1-x^2}\right) \right)+2\left(\frac{x}{2}log \left( \frac{1+x}{1-x} \right)-1\right)=1+\frac{1-x^2}{1-x^2}-2=0$$

Thus,
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$$L_{2} \left( {y}^{1}_{H}\right)=L_{2} \left( {y}^{2}_{H}\right)=0$$
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=(Referred from Wikipedia)Mathematical derivation of fictitious forces for understanding of Coriolis effect =

General derivation
Many problems require use of noninertial reference frames, for example, those involving satellites and particle accelerators. Figure 2 shows a particle with mass m and position vector xA(t) in a particular inertial frame A. Consider a non-inertial frame B whose origin relative to the inertial one is given by XAB(t). Let the position of the particle in frame B be xB(t). What is the force on the particle as expressed in the coordinate system of frame B?

To answer this question, let the coordinate axis in B be represented by unit vectors uj with j any of { 1, 2, 3 } for the three coordinate axes. Then

 * $$ \mathbf{x}_{B} = \sum_{j=1}^3 x_j\ \mathbf{u}_j \ . $$

The interpretation of this equation is that xB is the vector displacement of the particle as expressed in terms of the coordinates in frame B at time t. From frame A the particle is located at:

 * $$\mathbf{x}_A =\mathbf{X}_{AB} + \sum_{j=1}^3 x_j\ \mathbf{u}_j \ . $$

As an aside, the unit vectors { uj } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector XAB simply locates the origin of frame B relative to frame A, and so cannot include rotation of frame B.

Taking a time derivative, the velocity of the particle is:

 * $$ \frac {d \mathbf{x}_{A}}{dt} =\frac{d \mathbf{X}_{AB}}{dt}+ \underbrace{\sum_{j=1}^3 \frac{dx_j}{dt} \mathbf{u}_j}_{\mathbf{v}_B} + \sum_{j=1}^3 x_j\ \frac{d \mathbf{u}_j}{dt} \ . $$

The second term summation is the velocity of the particle, say vB as measured in frame B. That is:

 * $$ \frac {d \mathbf{x}_{A}}{dt} =\mathbf{v}_{AB}+ \mathbf{v}_B + \sum_{j=1}^3 x_j \ \frac{d \mathbf{u}_j}{dt} \ . $$

The interpretation of this equation is that the velocity of the particle seen by observers in frame A consists of what observers in frame B call the velocity, namely vB, plus two extra terms related to the rate of change of the frame-B coordinate axes. One of these is simply the velocity of the moving origin vAB. The other is a contribution to velocity due to the fact that different locations in the non-inertial frame have different apparent velocities due to rotation of the frame; a point seen from a rotating frame has a rotational component of velocity that is greater the further the point is from the origin.

To find the acceleration, another time differentiation provides:

 * $$ \frac {d^2 \mathbf{x}_{A}}{dt^2} = \mathbf{a}_{AB}+\frac {d\mathbf{v}_B}{dt} + \sum_{j=1}^3 \frac {dx_j}{dt} \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . $$

Using the same formula already used for the time derivative of xB, the velocity derivative on the right is:

 * $$\frac {d\mathbf{v}_B}{dt} =\sum_{j=1}^3 \frac{d v_j}{dt} \ \mathbf{u}_j+ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} =\mathbf{a}_B + \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} \ . $$

Consequently,
 * $$ \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_{AB}+\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . $$   (Eq. 1)

The interpretation of this equation is as follows: the acceleration of the particle in frame A consists of what observers in frame B call the particle acceleration aB, but in addition there are three acceleration terms related to the movement of the frame-B coordinate axes: one term related to the acceleration of the origin of frame B, namely aAB, and two terms related to rotation of frame B. Consequently, observers in B will see the particle motion as possessing "extra" acceleration, which they will attribute to "forces" acting on the particle, but which observers in A say are "fictitious" forces arising simply because observers in B do not recognize the non-inertial nature of frame B.

The factor of two in the Coriolis force arises from two equal contributions: (i) the apparent change of an inertially constant velocity with time because rotation makes the direction of the velocity seem to change (a dvB / dt term) and (ii) an apparent change in the velocity of an object when its position changes, putting it nearer to or further from the axis of rotation (the change in Σ xj $$\stackrel{\frac{d}{dt}}{}$$uj due to change in x j ).

To put matters in terms of forces, the accelerations are multiplied by the particle mass:

 * $$\mathbf{F}_A = \mathbf{F}_B + m\ \mathbf{a}_{AB}+ 2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . $$

The force observed in frame B, FB = m aB is related to the actual force on the particle, FA, by:

 * $$\mathbf{F}_B = \mathbf{F}_A + \mathbf{F}_{\mbox{fictitious}} \ . $$

where:

 * $$ \mathbf{F}_{\mbox{fictitious}} =-m\ \mathbf{a}_{AB} -2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} - m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . $$

Thus, we can solve problems in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treating Ffictitious as an additional force.

Below are a number of examples applying this result for fictitious forces. More examples can be found in the article on centrifugal force.

Rotating coordinate systems
A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Because such rotational motion is non-inertial, due to the acceleration present in any rotational motion, a fictitious force can always be invoked by using a rotational frame of reference. Despite this complication, the use of fictitious forces often simplifies the calculations involved.

To derive expressions for the fictitious forces, derivatives are needed for the apparent time rate of change of vectors that take into account time-variation of the coordinate axes. If the rotation of frame B is represented by a vector Ω pointed along the axis of rotation with orientation given by the right-hand rule, and with magnitude given by

 * $$ |\boldsymbol{\Omega} | = \frac {d \theta }{dt} = \omega (t) \, $$

then the time derivative of any of the three unit vectors describing frame B is:

 * $$ \frac {d \mathbf{u}_j (t)}{dt} = \boldsymbol{\Omega} \times \mathbf{u}_j (t) \, $$

and

 * $$\frac {d^2 \mathbf{u}_j (t)}{dt^2}= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j +\boldsymbol{\Omega} \times \frac{d \mathbf{u}_j (t)}{dt} $$ $$= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j+ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)\, $$

as is verified using the properties of the vector cross product. These derivative formulas now are applied to the relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with time-varying angular velocity ω ( t ). From the previous section, where subscript A refers to the inertial frame and B to the rotating frame, setting aAB = 0 to remove any translational acceleration, and focusing on only rotational properties (see Eq. 1):

 * $$ \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} $$ $$+ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ ,$$
 * $$\mathbf{a}_A=\mathbf{a}_B +\ 2\ \sum_{j=1}^3 v_j \boldsymbol{\Omega} \times \mathbf{u}_j (t)\ $$ $$+\   \sum_{j=1}^3 x_j \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j  \  + \sum_{j=1}^3 x_j \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)\ $$
 * $$=\mathbf{a}_B $$ $$+ 2\ \boldsymbol{\Omega} \times\sum_{j=1}^3 v_j \mathbf{u}_j (t) \  $$ $$+ \frac{d\boldsymbol{\Omega}}{dt} \times \sum_{j=1}^3 x_j \mathbf{u}_j  \   $$ $$+\ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \sum_{j=1}^3 x_j  \mathbf{u}_j (t) \right)\ .$$

Collecting terms, the result is the so-called acceleration transformation formula:

 * $$\mathbf{a}_A=\mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ $$ $$+ \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}_B  \  + \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ .$$

The physical acceleration aA due to what observers in the inertial frame A call real external forces on the object is, therefore, not simply the acceleration aB seen by observers in the rotational frame B, but has several additional geometric acceleration terms associated with the rotation of B. As seen in the rotational frame, the acceleration aB of the particle is given by rearrangement of the above equation as:

\mathbf{a}_{B} = \mathbf{a}_A - 2 \boldsymbol\Omega \times \mathbf{v}_{B} - \boldsymbol\Omega \times (\boldsymbol\Omega \times  \mathbf{x}_B )  - \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_B \. $$

The net force upon the object according to observers in the rotating frame is FB = m aB. If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional force Ffict is present, so the end result is FB = FA + Ffict. Thus, the fictitious force used by observers in B to get the correct behavior of the object from Newton's laws equals:

\mathbf{F}_{\mathrm{fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} - m \boldsymbol\Omega  \times (\boldsymbol\Omega \times \mathbf{x}_B ) $$ $$\ - m \frac{d \boldsymbol\Omega  }{dt} \times \mathbf{x}_B \. $$

Here, the first term is the Coriolis force, the second term is the centrifugal force, and the third term is the Euler force. When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero.

Reference http://en.wikipedia.org/w/index.php?title=Fictitious_force&action=submit

title: User:Egm6321.f10.team4.Yoon/HW1


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User:Egm6321.f10.team4.Yoon/HW2

= prob 3 ploting =

clear all; close all; clc;

%Constructing a Handle to an y_h1,y_h2; y_h1 = @(x) x; y_h2 = @(x) (x/2)*log((1+x)/(1-x))-1;

hold all fplot (y_h1,[-1.5 1.5],'b-'); fplot (y_h2, [-1.5 1.5],'r-');

xlabel('x'); ylabel('y');

title('Plots of y_1H and y_2H'); legend('y_1H = x', 'y_2H = x/2 *log(1+x)/(1-x)-1','location','north');

= - L1_ODE_VC Exactness =

= Problem 1-  =

From lecture page

Given
Verify that for a N1_ODE the following equation holds true.

$$\ {\rm{M}}\left( {x,y} \right) + {\rm{N}}\left( {x,y} \right)\frac = 0$$

Solution
Let (Eq.1-1) be a function of H(y) $$H(y)= {\rm{M}}\left( {x,y} \right) + {\rm{N}}\left( {x,y} \right)\frac$$

For the given general equation to be linear, it has to meet both of the following conditions, H is a dummy function and alpha and beta is an arbitrary constant.

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$$\displaystyle H(\alpha )=\alpha H $$ $$
 * $$\displaystyle (Eq. 1-2)
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$$\displaystyle H(\alpha+\beta)=H(\alpha)+H(\beta) $$ $$
 * $$\displaystyle (Eq. 1-3)
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By the first condition of Linearity,
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$$\displaystyle H(\alpha )= {\rm{M}}\left( {x,\alpha y} \right) + {\rm{N}}\left( {x,\alpha y} \right)(\alpha y)' $$

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$$\displaystyle \alpha H = \alpha[{\rm{M}}\left( {x,y} \right) + {\rm{N}}\left( {x,y} \right)y'] $$ Thus,
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$$\displaystyle H(\alpha ) \neq \alpha H' $$
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By the second condition of Linearity,
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$$\displaystyle H(\alpha+\beta)={\rm{M}}\left( {x,\alpha + \beta} \right) + {\rm{N}}\left( {x, \alpha + \beta} \right) \left(\alpha + \beta \right)' $$
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$$\displaystyle H(\alpha)+H(\beta)={\rm{M}}\left( {x,\alpha} \right) + {\rm{N}}\left( {x, \alpha} \right)(\alpha)'+{\rm{M}}\left( {x,\beta} \right) + {\rm{N}}\left( {x, \beta} \right)(\beta)' $$
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Thus,
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$$\displaystyle H(\alpha+\beta) \neq H(\alpha)+H(\beta) $$
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Since the given equation doesn't satisfy neither of the linearity conditions, the given general N1_ODE_VC is non-linear.

= Problem 6 - Verify condition of Exactness =

Given

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$$\displaystyle y(x) = \sin ^{ - 1} (k - 15x^5 ) $$     (Eq 6.1)
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Verify the given equation above satisfies the following First condition of exactness,

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$$\displaystyle M\left(x,y\right)+N(x,y)y'=0 $$
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Solution

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$$\displaystyle M(x,y):={\phi}_{x}(x,y)=\frac{\partial \phi(x,y)}{\partial x} $$
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$$\displaystyle

N(x,y):={\phi}_{y}(x,y)=\frac{\partial \phi(x,y)}{\partial y} $$
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From the (Eq 6.1),

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$$\displaystyle 15x^5+\mathrm{sin}\left[y\left(x\right)\right]=k =: \phi(x,y) $$     (Eq 6.)
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Total differential of function $$\phi\left(x,y\right)$$,

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$$\displaystyle d\phi\left(x,y\right)=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy $$     (Eq 6.)
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As$$\phi\left(x,y\right)=k=const$$, then $$ d\mathbf{\phi}=0 $$, We can rewrite the (Eq 6.),

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$$\displaystyle \frac{\partial \phi}{\partial x}+\frac{\partial \phi}{\partial y}\frac{dy}{dx}=0 $$     (Eq 6.)
 *
 * }

For the given problem,
 * {| style="width:100%" border="0" align="left"

$$\displaystyle {\phi}_{x}(x,y)=\frac{\partial \phi(x,y)}{\partial x} = 75x^4 $$     (Eq 6.)
 *
 * }

 * {| style="width:100%" border="0" align="left"

$$\displaystyle {\phi}_{y}(x,y)=\frac{\partial \phi(x,y)}{\partial y} = \mathrm{cos}y $$     (Eq 6.)
 *
 * }

So, we can verify that the eqn(6.1) could be converted to the form of the first condition of exactness(6.2).

For the given problem,
 * {| style="width:100%" border="0" align="left"

$$\displaystyle \underbrace{\left(75x^4\right)}_{M\left(x,y\right)}+\underbrace{\left(\mathrm{cos}y\right)}_{N\left(x,y\right)}\cdot y'=0 $$     (Eq 6.)
 *
 * }

Given

 * {| style="width:100%" border="0" align="left"

\frac{1}{3} x^3 y^4 y' + \left(5x^3 + 2 \right) \left(\frac{1}{5} y^5 \right) = 0 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

Find
Show that Eq.(1) is an exact nonlinear, first order ODE(N1_ODE_VC).

Solution
with rearranging the given Eq.(1) by reduction order of y
 * $$ \underbrace{(5x^{3}+2)\left(\frac{1}{5}y^{5}\right)}_{M(x,y)} + \underbrace{\left(\frac{1}{3}x^{3}y^{4}\right)}_{N(x,y)}y'=0$$

We can find the first exactness condition is met by defining M(x),N(x). However,
 * {| style="width:100%" border="0" align="left"

M(x,y) := \left(5x^3 + 2 \right) \left(\frac{1}{5} y^5 \right) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

N(x,y) := \frac{1}{3} x^3 y^4 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 3)
 * }
 * }

To satisfy the second condition of exactness, My = Nx
 * $$M_y = 5x^3y^4+2y^4 \neq x^2y^4 = N_x$$

To find an integrating factor, deviding the (Eq. 1) by :$$\frac{1}{3}x^3y^4$$ to make the coefficient of Y' to 1
 * {| style="width:100%" border="0" align="left"

y' +\frac{3}{x^3y^4}(5x^{3}+2)\left(\frac{1}{5}y^{5}\right) = 0 $$

\underbrace{1}_{N}\cdot y' + \underbrace{\left(3 + \frac{6}{5x^3}\right)}_{a(x)}y = 0 $$

Now we will make use of integrating factor knowledge:

\Phi(x,y) = \exp\int^{x}a(s)ds = \exp\int^{x}\left(3 + \frac{6}{5s^3}\right)ds = \exp\left(3x - \frac{3}{5x^2}\right) $$

multiplying Φ(x,y) through to find,
 * $$\bar{M}(x,y)=M(x,y) \cdot \Phi (x,y)$$
 * $$\bar{N}(x,y)=N(x,y) \cdot \Phi (x,y)$$

so to meet the second exactness condition,
 * $$\bar{N}_{x}=\bar{M}_{y}$$

Substituting and differenciating,
 * $$\bar{M}y=exp \left(3x-\frac3{5x^2}\right)\cdot \left(3+\frac6{5x^3}\right)$$
 * $$\bar{N}x=exp \left(3x-\frac3{5x^2}\right)\cdot \left(3+\frac6{5x^3}\right)$$,

Thus above is satisfying $$\bar{N}x=\bar{M}y$$, (Eq. 1) is an exact N1_ODE_VC

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

title: User:Egm6321.f10.team4.Yoon/HW2


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text:
Dot product

The dot product, also commonly known as the "inner product", or, less commonly, the "scalar product", is a number associated with a pair of vectors. It figures prominently in many problems in physics, and variants of it appear in an enormous number of mathematical areas.

Geometric Definition
It is defined geometrically as the product of the lengths of the two vectors times the cosine of the angle between them. This definition is used in 2 dimensions (plane geometry) or 3 dimensions (solid geometry and physics.) The dot product is written with a raised dot between the vectors,
 * $$\vec{A} \cdot \vec{B} = AB\cos\theta$$,

where $$\theta$$ is the angle between the two vectors. Following a common convention, the magnitudes have been written as the variable without the vector symbol: $$A\equiv ||\vec A||$$, and $$B\equiv ||\vec B||$$.

A few properties are immediately apparent.
 * It is symmetric: $$\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\,$$.
 * It is linear in each argument: $$(\lambda\vec{A}) \cdot \vec{B} = \vec{A} \cdot (\lambda\vec{B}) = \lambda (\vec{A} \cdot \vec{B})\,$$.
 * If the two vectors are orthogonal (perpendicular) to each other, their dot product is zero. This is because the cosine of 90° is zero.
 * The length of a vector is the square root of the dot product of the vector with itself. This is because the angle is 0, so the cosine is 1.
 * $$\|\vec{A}\| = \sqrt{\vec{A} \cdot \vec{A}}\,$$
 * (The double absolute value sign, shown above, is known as the norm. It can be thought of as a generalization of the absolute value of a number.  It is used in many places in mathematics.  For vectors, it is the length.)

Algebraic definition
It turns out that, given the components of vectors in a Cartesian (Euclidean) vector space, there is an extremely simple way to calculate the dot product. It is the sum of the pairwise products of the components of the vectors:
 * $$\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\,$$.

or, depending on what notation you prefer:
 * $$\vec{A} \cdot \vec{B} = A_1B_1 + A_2B_2 + A_3B_3 = \sum_{i=1}^3 A_iB_i\,$$.

This definition is also used in 2 dimensions, with only two terms in the addition.

 * It needs to be emphasized that this formula only works if the components of the vectors are described in Cartesian space. In other coordinate systems, it may be necessary to use other, somewhat more complicated formulas.

The equivalence of this algebraic definition and the earlier geometric one is extremely important. Among many other things, it lets us calculate the angle between two vectors, given their components. Calculate the lengths of the vectors as the square root of the dot product of each vector with itself, calculate the dot product of the two vectors, divide by the lengths, and take the inverse cosine:
 * $$\theta = \cos^{-1} \left(\frac{A_xB_x + A_yB_y + A_zB_z}{\|A\| \|B\|}\right)\,$$.

Using the algebraic definition, one can see that the dot product is additive in each argument:
 * $$\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\,$$.
 * $$(\vec{A} + \vec{B}) \cdot \vec{C} = \vec{A} \cdot \vec{C} + \vec{B} \cdot \vec{C}\,$$.

something which is not so easy to see from the geometric definition.

Proof of Equivalence of the Two Definitions
If one is good at geometric visualization in 3 dimensions, and has a good understanding of the Euclidean rotation group, one can rotate the Cartesian coordinate system around until the two vectors lie in the x-y plane, and the first vector lies along the x axis.
 * This requires knowing that any rotation around a coordinate axis is a transformation by an orthogonal matrix, that the sum of the pairwise products of components is unchanged by an orthogonal transformation, and that the two vectors can be rotated into the required position by a combination of rotations around coordinate axes.
 * Once this is done, it becomes a problem in 2 dimensions.

$$B_x\,$$ is the projection of $$\vec{B}\,$$ onto the x axis, which is also the length of $$\vec{B}\,$$ times the cosine of $$\theta\,$$.
 * $$B_x = \|B\| \cos\theta\,$$
 * $$A_x = \|A\|\,$$
 * $$A_y = 0\,$$
 * $$A_xB_x + A_yB_y = A_xB_x = \|A\| \|B\| \cos\theta\,$$

Another way to prove this is to prove it first for the norm of a vector, and then use the law of cosines.

We first need to show that the geometric length of $$\vec{A}\,$$ is $$\sqrt{A_x^2 + A_y^2 + A_z^2}\,$$

Whatever coordinate system we are in, the vector $$\vec{A}\,$$ runs from one corner to the opposite corner of a rectangular box with sides $$A_x\,$$, $$A_y\,$$, and $$A_z\,$$.

Two applications of Pythagoras' theorem suffice to prove the required result. The length of a vector is the square root of the sum of the squares of its components.

Next, whatever coordinate system we are in, form the plane containing $$\vec{A}\,$$ and $$\vec{B}\,$$, and use plane geometry. Let $$\vec{C} = \vec{A} - \vec{B}\,$$. That is, $$\vec{C}\,$$ runs from the tip of $$\vec{B}\,$$ to the tip of $$\vec{A}\,$$. Form the triangle with those vectors as sides.

By the law of cosines for a triangle:
 * $$\|C\|^2 = \|A\|^2 + \|B\|^2 - 2 \|A\|\ \|B\| \cos\theta\,$$

Since we have established that the algebraic definition works for calculating norms:
 * $$C_x^2 + C_y^2 + C_z^2 = A_x^2 + A_y^2 + A_z^2 + B_x^2 + B_y^2 + B_z^2 - 2 \|A\| \|B\| \cos\theta\,$$

Since $$\vec{C} = \vec{A} - \vec{B}\,$$:
 * $$A_x^2 + B_x^2 - 2 A_xB_x + A_y^2 + B_y^2 - 2 A_yB_y + A_z^2 + B_z^2 - 2 A_zB_z = A_x^2 + A_y^2 + A_z^2 + B_x^2 + B_y^2 + B_z^2 - 2 \|A\| \|B\| \cos\theta\,$$

so
 * $$- 2 A_xB_x - 2 A_yB_y - 2 A_zB_z = - 2 \|A\| \|B\| \cos\theta\,$$

so
 * $$A_xB_x + A_yB_y + A_zB_z = \|A\| \|B\| \cos\theta\,$$

title: Dot product


text:
Motivation and emotion/Textbook/Motivation and emotion/Animals

Introduction
Psychology is a fascinating topic and animal psychology is almost a new science for researchers and pet owners alike. This chapter is about theory and research on animal learning and  training concepts,  animal communication, with discussion about current training methods. Motivation and emotion are areas that researchers have explored for decades and training concepts have evolved as a result of this exciting new field of animal research. The relationships between animals and people have evolved over centuries and the impact of animal domestication on man's progress is considerable. Hopefully both scientists, pet owners and psychology students will gain some new insights.

This chapter will attempt to unravel some of the mysteries of animal behaviour and how it relates to emotions and motivation. Have you ever wondered why a dog will bond to one owner, while another dog will attack its keeper? Why do elephants stand quietly at a circus with a small chain attached to its leg? How are dolphins, seals and killer whales trained? Why are most horses reluctant to get on a horse float? Can a reptile be trained?

Motivation and emotion are underpinned by learning theory, behaviourism, evolutionary and biological factors. Drive instincts ensure survival and explain emotional states. Of the many theorists and researchers who have contributed to the study in this field, this chapter will exclusively deal with those involved in animal behaviour. It is widely acknowledged that when animals are born, instinct  and  learned behaviour have enabled animals to adapt to their environment, be it in the wild or in a domestic situation.

Do they really think? Animal cognition explained
Intelligence in animals such as dolphins could be related to brain size since research has shown that dolphins, whales and elephants all have large brain sizes. It is claimed to be a result of mutual dependence based on external threats from predators and con-specific groups(Connor, 2007). There are three “peaks” of brain size evolution in mammals and dolphins have brain sizes similar to humans, (Morisaka, 2009).

Comparisons are drawn between different mammals and their brain size, indicating that body size to brain size ratio is not a strong factor in intelligence. Humans have the largest brain relative to body size with the added benefit of language. Language is one thing that defines humans from animals, (Connor, 2007) however it is acknowledged that dolphins also have their own language of squeaks and whistles. It strongly suggests that they have complex communication skills which could be correlated to their level of intelligence.

Mammals with larger brains may have better cognitive abilities for complex environments, which could be explained by the evolutionary expansion of the brain and development of the forebrain. (Ratcliffe, Fenton, & Shettleworth, 2006) It is claimed that the neural substrate in the neocortex is responsible for risk assessment and decision making. The hippocampus has also been determined as a vital part of memory retention and spatial knowledge. The size of the hippocampus and neocortex is directly proportional to the cognitive abilities of birds and mammals and relative to their body size. It stands to reason that mammals with larger brains and larger parts of the brain are better equipped for predator evasion (Ratcliffe et al., 2006).

Studies on animal cognition have been carried out with dogs, cats (Leblanc & Duncan, 2007) monkeys, rats and birds. (Cook, 1993) Honey bees have been studied to determine how they cognitively make sense of their spatial world as the need to navigate to and from the hive regularly. (Cheng, Srinivasan, & Zhang, 1999) It is claimed that many animals both vertebrate and invertebrate can find their way back home or back to a previous destination, partly it is claimed due to spatial information processing and recognition of landmarks. Relatively few studies have been done with horses (Cook, 1993). In humans it is known that the left side of the brain generally controls the analytical side and the right half involving emotion.(McGreevy, 2004) Veterinarian Dr. Paul McGreevy suggests that horses are similar.

It is a fascinating comparison between animal and human cognition. Robert Cook (1993) posits that not enough research in this area is done with different species of animal, he states that using a pigeon to represent some 9000 species of birds is fraught with peril. To understand the aspects of the functional significance of cognitive animal intelligence, more study in the areas of distribution, ecology and spatial knowledge need to be studied in depth. How do rats gain spatial information through a maze? Is it as simple as the reward at the end giving incentive in order to navigate it faster each time it passes through? How does avian spatial ability differ from non human animals and man?

To find out, Cook’s (1993) study using series of complex line drawings and patterns found that both pigeons and humans can discriminate feature displays better than conjunctive displays. The similarities between how pigeons discriminated between the pattern shapes were similar in man, finding the density of the dots dictated how accurately the pictures were targeted. This visual discrimination is understandable in birds given their high visual acuity, necessary for finding food (birds of prey) or flying away from danger. It also supports evidence that birds process cognitive information to great detail, as do other species.

Another indicator of complex animal cognition is that of imitation as defined by Thorndike, (Byrne, 1999) as “learning to do an act by seeing it done” or copying another simply by observational learning. Further refined by M.Tomasello as the concept of emulation, (Byrne, 1999) where an example may be an individual monkey who may see another monkey accidentally open a coconut by dropping it from a great height, breaking it open, thus the obvious reward is food. It is posited that the cells of the mirror neurons in monkey cortex area of the brain all respond equally to the same actions such as mutual grooming and opening peanuts.

The idea of imitation has been challenged, (Byrne, 1999) however the current evidence is hard to ignore when there are examples of such behaviour in birds such as naïve tits (birds) that learn how to pick the caps off bottles. Another is crows that are reported to have learned how to flip cane toads on their backs and peck out their stomachs, as it’s the only area on the cane toad that is not deadly poisonous. These complex social processes have developed from selective pressures of survival of the species.(Boysen & Himes, 1999) A study using chimpanzees raised by humans, some with natural mothers and some with human children on imitation tasks had fascinating results. On the simple and complex tasks, the mother reared chimpanzees were the poorest performers. However, on the final task which was an imitation task with a 48 hour delay, the monkeys raised by humans actually outperformed the mother reared chimpanzees and human children. (Boysen & Himes, 1999) Capuchin monkeys have also been investigated on tool use skills and have demonstrated understanding of the causal effects, compelling evidence of animal cognition.

Chimpanzees have also been claimed to learn by imitation how to use pieces of plants to dig termites out of nests, (Byrne, 1999) and use stones to crack open nuts. Rats have also been known to copy each other, black rats in Jerusalem have learned how to open pine cones in a systematic way which is claimed to not be natural to that species. Similar to squirrels, the rats have learned how to break off the cones carefully then take them to another branch and meticulously peel them back to get the seeds inside. It was discovered through cross breeding to not have genetic origin, but is social in nature. It was stated that this skill does not come naturally to the rats when presented with unopened cones, (Byrne, 1999) even if they were starving. It was also claimed that the rats could be taught by man how to open the cones, strengthening the argument for emulation.

Do animals have personalities?
More recently, the field of animal personality been more intensively studied and it has a direct relationship to cognition, or their awareness. It is now becoming recognized more widely that animals have individual personalities just the same as different breeds have certain traits. Sonia Cavigelli’s (2005) longitudinal study used animal subjects, specifically due to the recognition that animals have different personalities. She wanted to find out if there was a relationship between personality, physiology and health in individuals and because some animals have a short lifespan, they were perfect for such a study. She wanted to discover the impact individual differences made to disease susceptibility that could be linked to similar human studies.

She defines personality as a set of inter-related traits that are stable within an individual across time and across situations (Cavigelli, 2005). Her study concentrated on the relationship between personality and health, but there is not a lot of research on specific animal personalities in the area of animal learning. Gosling & John’s (1999) study examined the “five factors”, or the five personality factors found in humans and identified certain personality traits across species of animals. (Cavigelli, 2005) This suggests that some personality traits are genetic and research in this area of animal personality could reveal common elements of personality also found in humans.

A more recent study on social cueing in dogs fully embraced the concept of personality in dogs, so much so that part of the research they did included a dog personality questionnaire. They recognize that dogs follow social cues from humans such as pointing to a hidden object. They found that dogs outperformed primates, wolves and goats on similar tasks which are of great interest to cognitive scientists, (Clotfelter & Hollis, 2008) The suggestion that domestic animals have superior cognition suggests the idea that dogs have a “theory of mind”, which is of value in human-animal communication. One dog training system includes dog personality tests into its training regime which may represent the recognition of dog cognition. See http://www.volhard.com/pages/canine-personality-profile.php

Adaptability and behavioural flexibility are requirements for animals to succeed in any environment they encounter, (Dingemanse & Réale, 2005). Dingemanse (2005) did a study on aspects of individual differences in animals with case studies in different environments. He wanted to see how similar personalities affected how they coped in different situations. He found that hand reared great tits (birds)reactions were comparable in their individual reactions to rats on being presented with novel stimuli. (Dingemanse & Réale, 2005) He did another case study on big horn sheep, where he found certain sheep would enter an enclosed yard baited with salt and others wouldn’t, supposedly measuring the trait of boldness. He discussed boldness in fish from another experiment, indicating their aggressiveness enabled them to find food against competition, thus increasing their chances to survive and pass on genetic traits.

Probably the most fascinating of the studies discussed was that of black rhinoceros (Diceros bicornis) in 24 zoos. The females kept in captivity that were observed to display less stereotypical behaviour such as mouthiness and chasing had the highest reproductive success. Also, those that showed fear, docility and displayed patrolling activity were not significantly related to reproductive success. Cheetahs in the same type of multi zoo study had similar results where non breeders were more fearful than breeders. (Dingemanse & Réale, 2005) The findings of these studies suggest that personality characteristics in different species enable animals overall survival, due to aggressiveness or risk taking behaviour.

The value of studies of personality in animals cannot be underestimated, especially in the field of larger domestic animals such as horses and cattle. Scientific studies on horse personality would be extremely fascinating as it could answer many questions about equine cognition. The literature reveals that animals appear to have individual personalities,(Carere & Eens, 2005) so why not horses? Understanding individual equine personality could be of value in the horse training industry to train the horses more effectively depending on their innate characteristics. More about this type of training is discussed in the supplementary section.

Classical and operant conditioning, how animals learn
How animals learn underpins what motivates them and suggests they possess awareness. The need to find food facilitates learning, from opening nuts to killing a large beast. How they learn is explained by the following theories and research.The phenomenon of learned helplessness will be covered and imprinting learning during critical learning periods. The definitions of different types of reinforcement, conditioned and unconditioned stimulus and how they are presented for the purposes of training or shaping behaviour will be discussed in this section.

Ivan Pavlov was a Russian professor who was awarded a Nobel Prize in 1904 for his groundbreaking work on digestive tracts in dogs which eventually led to his discovery of conditioned reflexes (Marks, 2004). He observed what happened in the digestive tract when he rang a bell then fed the dogs as a reward, leading him to discover how the cerebral cortex responded to certain reflexes (Ivan Petrovich Pavlov, 2009). The conditioned reflex, now known as classical conditioning was that the dogs started salivating when he rang a bell, because they had been conditioned to do so by food rewards.

Pavlov discovered how conditioned and unconditioned responses worked. (Powell et al., 2005) An unconditioned response would be a response elicited prior to conditioning, where the dogs would salivate when they tasted their food and since this occurs naturally it is an unconditioned response, the food is called an unconditioned stimulus.

A conditioned response on the other hand is a trained response. An example would be the sound of a bell when presented by itself to the dog. It elicits no response so it is a neutral stimulus as no salivation occurs. Once the sound of the bell and the giving of food are repeatedly paired together, the sound of the bell now makes the dog salivate. Once the pairing association has been made by the dog, it will salivate on the sound of the bell alone, so the dog is giving a conditioned response. The bell is now the conditioned stimulus. Pavlov also found that spontaneous recovery was not a process of unlearning the conditioning, but was new learning taking place, so the dog learns to inhibit the response to the bell during extinction.

Disinhibition, is another phenomenon defined as when a response suddenly recovers during extinction when a different stimulus is introduced. Pavlov discovered during his research on experimental neurosis that under certain experimental conditions, his dogs displayed extremely neurotic behaviour such as anxiety and in extreme cases catatonia. (Powell et al., 2005) Pavlov's work heralded the beginning of the field of behaviourism.

Prior to his findings, John B. Watson (Behaviorism.2009) believed that behaviour was simply a response to something in the environment, assuming no mental processes occurred. He discovered that food effectively motivated certain responses with positive reinforcement. Watson further developed and expanded research based on Pavlov’s work on Behaviorism and wanted to study the emotional aspects of classical conditioning (Rilling, 2000). He wanted to find out more about fear in children and believed all behaviour was based on reflex conditioning from an environmental stimulus. He discovered he could condition certain emotions such as fear into children and developed his learning theory.

B.F. Skinner expanded on behaviourism in the 1930’s, he believed external events could motivate behaviour. Using punishment and reward (withholding food) to train pigeons and rats to perform certain tasks (Strickler, 2006) he got birds to peck for food and rats to press a lever to get food, this is now known as operant conditioning. (Iverson, 1992) In contrast to Edward Thorndike's earlier research on  learning by trial and error, also known as the  law of effect, Skinner collected data which expanded considerably on what was discovered. He developed reinforcement  schedules and developed procedures to shape behaviour by  successive approximation. He didn't realize the conditions he elicited could be used to change emotional states in scared animals. Skinner developed the Skinner Box (Measured learning.1992)as part of his animal studies(Epstein, 1995).

For learning to occur, there must first be a stimulus. There are two types of stimulus, appetitive and aversive. An appetitive stimulus is actively sought out, such as food when hungry and an aversive stimulus is that which is actively avoided such as an electric shock. Contiguity and contingency refers to the closeness of the stimuli, such as an event that happens close together in time and it is defined as temporal contiguity. (Powell et al., 2005) Events happening in close proximity to each other in space are defined as spatial contiguity and those events that have a predictive nature between the two are known as contingency. These laws all relate to conditioning and they explain the relationships between stimulus and habituation, conditioned and unconditioned responses which underpin classical and operant conditioning.

Note. Adapted from "Introduction to Learning and Behaviour" by R. Powell, D. Symbaluk and S. MacDonald, 2004.

In classical conditioning, habituation is defined as a decrease in certain behaviours as a result of repeated exposure to a certain stimulus, for example low background noises in a new environment. We may find ourselves in an area with busy traffic noise but after a while we gradually become less aware of it and are habituated to it.

Desensitization is where the stimulus is presented repeatedly until there it no longer elicits a response, similar to habituation Sensitization is when a stimulus is presented repeatedly in such a way that we become very sensitive to it. Being near a railway where the rattle and roar of the trains initially startles you, but after a while they don’t bother you any more as you become desensitized to the noise.

An appetitive (unconditioned) stimulus is usually associated with a pleasant experience while the opposite is and an aversive (unconditioned) stimulus is very unpleasant so is avoided. (Powell et al., 2005) An example of this would be a bull that touches an electric fence and receives an immediate shock. Aversive conditioning occurs more quickly, reflecting the innate mechanisms of survival that says we must quickly learn to avoid that which causes harm or pain (Powell et al., 2005).

The strengthening of an associated repeated pairing of unconditioned stimuli is known as acquisition. The conditioned response can also be weakened considerably or eliminated by repeated pairings of the conditioned stimulus without the conditioned stimulus. However responses can spontaneously recover if there is a rest period between extinction and the next conditioned response. It is weaker than the original response and does not last permanently. (Powell et al., 2005)

Classical conditioning techniques are used on humans to treat phobias and given what the literature indicates about animal personalities, classical conditioning can also be used to treat phobias in animals. Some equine trainers already use classical conditioning using systematic desensitization, flooding, habituation and successive approximation or approach retreat.

Operant conditioning is another type of learning where the probability of a future behaviour occurs as a result of direct consequences (Powell et al., 2005). Edward Thorndike had an interest in animal behaviour and he formulated what is known as the law of effect. It states that behaviours leading to a satisfactory state of affairs are strengthened or solidified and unsatisfactory behaviours are weakened or eliminated (Powell et al., 2005). Skinner as previously discussed further expanded on this work on how consequences facilitate learning with his Skinner box and experimental research with rats and pigeons.

Reinforcement or punishment could be defined as the consequence. The four contingencies of reinforcement are positive reinforcement, negative reinforcement, positive punishment and negative punishment.

Positive reinforcement means something pleasant is added to encourage the desired behaviour as the picture of the rat illustrates. He has figured out how to self administer morphine as a result of pressing a lever. Another example would be a crocodile that learns to jump up out of the water to eat a chicken. Crocodiles are known to jump up out of the water for food especially when tourist operators regularly feed them as part of the entertainment, the behaviours are positively reinforced.[]

Negative reinforcement something unpleasant is subtracted to encourage the desired behaviour, in the case of the rat he can stop an electric shock he is receiving by pressing the lever. (Powell, Symbaluk, & MacDonald, 2005) The behaviour (pressing the lever) increases in strength because the rat removes something unpleasant (the electric shock).

Positive punishment means something unpleasant is added to discourage the undesired behaviour. An example would be an animal that puts its nose on an electric fence, every time he touches it he gets an electric shock, so he learns to not touch it. Negative punishment means something pleasant is subtracted to discourage undesired behaviour. The example given in Powell (2005), is that of staying out past curfew and losing car privileges as a result.

Shaping is the next stage of operant conditioning. Karen Pryor (Powell et al., 2005) who is a clicker training advocate uses this method to train dogs with a clicker as a secondary reinforcement. She suggests that in order to train a dog to catch a ball, first take the ball from the dog and immediately return it, which reinforces the behaviour. Then in successive stages, gradually go further away from the dog before returning the ball until he first has to jump a little to catch it, then eventually he has to chase it down to catch it. In conjunction with the use of the clicker the behaviour is reinforced or shaped. Clicker training will be explained in more detail in a later next section. These concepts give the basis of motivation in animals and they are invaluable in terms of animal learning and training methods.

Instinct, drive reduction theory and learned helplessness
Instinct could be defined as that which is genetically coded at birth and learned behaviour is that which takes a little longer to develop. (Mills & Nankervis, 1999) The two go hand in hand since what an animal is genetically endowed with will be enhanced by its environment and experiences. An example of instinctive behaviours in mammals could be the initial standing up at birth, followed by identifying mother and seeking milk and nursing. Later on, the instincts are further reinforced and enhanced through environmental experience, nature and nurture together.

A part of instinctive behaviours also includes staying with the herd, finding water, food and rest. When any of these are needed, an animal will be driven to actively seek it out until he finds enough to reach homeostasis. If he is thirsty and water is in short supply and he goes thirsty for long enough, the drive to find water will motivate him even more the longer he is without it until he finds it. When he finds water and drinks his fill, his physiological needs are met so the drive is reduced again. Clark Hull had a view on drive reduction and reinforcement and his theoretical approach was known as the Premack principle. (Powell, Symbaluk, & MacDonald, 2005) According to Hull, all reinforcement is associated with drive reduction in one way or another. The Premack principle principally claims that high-probability behaviours can be used to reinforce low-probability behaviours. An example of the Premack principle would be first do your homework, then go and play. It focuses on the underlying motivation of the behaviour rather than just the stimulus presented as the reinforcement.

Learned helplessness is a state that could be defined as that of the animal seeking to avoid pain or discomfort from an unpleasant stimulus (Curtis & Stricklin, 1991), but never finding a way out, as a result they become dull and listless and just give up. It could also be defined as a decrement in learning ability that results from repeated exposure to uncontrollable aversive events. (Powell, Symbaluk, & MacDonald, 2005). An example would be a baby elephant who learns he cannot escape the leg chain no matter how hard he tries. He remembers this for life.

Imprinting
Imprinting is a naturally occurring phenomenon that modern trainers use to reduce fear in their animals. It occurs at birth and researcher Karl Lorenz experimented with imprinting geese and demonstrated how geese and ducklings will imprint on the first thing they see after they are born. [] The video has no audio but demonstrates how Lorenz gets the geese to imprint onto him and follow him everywhere (2 min). Animals of different species will also imprint on other animals as the next video shows,[].The explanation of this phenomenon is in that the instinct of a precocial prey species is to bond to whatever it first sees at birth, (since it is usually the mother) for protection from predators. The interesting thing Lorenz also found with geese that imprinted on foster mothers was that they also imprinted sexually and tried to court and mate with species the same as the foster mother when they reached sexual maturity. (Manning, 1967) Dr Robert Miller is a veterinarian who advocates imprint training and uses it to gentle foals and other wild precocial species (Miller, 2007). In his book “Natural Horsemanship Explained” some pictures are of him imprinting zebra foals and even a rhinoceros. He also authored a book titled “Imprint Training of the Newborn Foal” where he explains the process that he uses and how effective it is from a veterinarians’ point of view. The main point to note is that animals have a sensitive or critical phase at or soon after birth and it varies between predators and prey animals. When predators are born, they are born blind and imprint or learn best a few weeks after birth. Prey animals or precocial species must be immediately equipped to be perceptive to danger and be able to flee from birth, so imprinting to what is usually the mother is vital for their survival. The level of motivation an animal has will also affect their rate of learning, (Manning, 1967) and it is claimed that animals vary in what motivates them the most. Some animals are motivated by food, others by simply being left alone. It is also claimed by the author that the extent of the animal’s learning ability is roughly proportional to the extent of its cerebral hemispheres since they are involved in memory. Precocial species are neurologically mature at birth and imprinting is valuable for training purposes since the fear instinct is greatly reduced, (Miller, 1999). Imprinting foals at birth has an indelible effect (McGreevy, 2004) and is highly efficient due to the speed of learning, making them easier, quieter and less dangerous to handle when they are fully grown. This video shows Robert Miller demonstrating the excellent results on an imprinted Mule foal[](video approx 8min). The next section explores emotions further.

Emotion in animals
This section discusses current theory and research on emotions in animals and considers implications for animal welfare. Arguably the best example of emotions in animals would be the reactions of baby Rhesus monkeys on separation from their mothers as was done by Harry Harlow in the 1960’s. As documented in the book “Early Learning and Early Experience”, (Sluckin, 1971) the author gives a detailed account of one experiment where Harlow spoke about what happened when the infant monkeys were initially separated from their mothers. He described them as being disoriented, screaming and crying immediately after separation and he indicated the intensity of the crying being significantly higher immediately following separation than before. Following separation once they settled down, they displayed depressive symptoms indicating severe emotional distress, such as low activity, little or no playing and occasional crying, similar to despair seen in children who are separated from their mothers. These reactions could be argued to be exactly the same emotions that humans experience.

John Bowlby was a London psychoanalyst who's research concentrated on attachment theory in the early 1960’s (Van, LeRoy, & Van, 2008). It is claimed in the literature that Bowlby’s work influenced the study of Ethology (van, van, & van Ijzendoorn, 2007) and the mother-child relationship for scientists such as Tinberger, Hinde and Harlow. Harry Harlow was an American animal psychologist who used Rhesus monkeys for his research. It could be said that between Bowlby and Harlow, the discoveries the two of them made around attachment and the effects of parent - child separation were groundbreaking and underpin most of what we know today about the effects of early separation, grief and loss. While Bowlby primarily concentrated on the relationships between children and their mothers while working at the Tavistock clinic in London (Van et al., 2008), Harlow initially studied the ontology of learning and motivation. He became more interested in the emotions of maternal affection (Van et al., 2008) as he observed the behaviour of his monkeys and conducted a series of experiments on the effects of separation.

They discovered psychological emotional distress such as depression, despair, stereotypical behaviour, separation anxiety (Bretherton, 1997) were caused by broken attachments. The video is about how the brain alters its structure and functioning in response to social situations. The impact of different stimuli on human and animal brains, from the effect of human touch on premature babies to the effect of social status on the health of baboons. (runs approx 27min)

 * http://www.learner.org/discoveringpsychology/04/e04expand.html#

A documentary screened in 2006 by ABC TV Australia, “Monkey Love”,  gave great insight and a fascinating account of the man and his work, although confrontational. He wanted to find out if monkeys were motivated by food or comfort of a mother by separating babies from mother at birth and placing them with a cloth mother, see video []. He found they preferred the cloth mother []emphasising the importance of maternal care in animals that can be related back to human children. []There is video footage of them displaying fear [] aggression, self mutilation and complete despair occurred in infant monkeys after being left in total darkness for up to 2 years.

http://www.abc.net.au/4corners/content/2006/s1658576.htm

In another article, (Bekoff, 2000) the author makes an argument that followers of Skinner’s work believe that animals are simply machines that respond to stimuli. There is little evidence that emotions were never considered, since they are difficult to measure and the argument is losing its momentum according to new evidence claimed by some researchers. (Carmichael, Reno, & Shenfeld, 2003) Although pet owners may interpret certain animal actions as love, it cannot be denied that feelings of fear, despair, jealousy and love in animals exist. Animals respond to kind treatment, good food and having their physiological needs met.

It is posited in the book “Understanding Motivation and Emotion”, (Reeve, 2009) that anticipation of some event can cause an emotion to occur. As one example, trying to get a horse into a horse float can be a challenging prospect for anyone who has ever owned a horse. How can a person who cannot out-muscle a huge, scared, prey animal to get on a small enclosed, dark box on wheels without killing themselves or the horse in the process? There are several different and somewhat controversial methods to resolve this problem which will be addressed in a later section while attempting to explain the emotional aspects.

What emotions do animals experience?
Agricultural industries are claimed (Curtis & Stricklin, 1991) to be slow to accept the notion of animal awareness according to Curtis & Stricklin. They suggested that cognition and awareness in animals has a direct physiological effect on their level of production due to stress caused by their direct experience. For example, does a dairy cow experience fear or stress in the milking barn?

There is a direct correlation between lowered productivity and six causes of stress found in chicks, they are ammonia, beak trimming, coccidiosis, electric shock, heat and noise. (Curtis & Stricklin, 1991) It is widely recognized that how an animal really feels is difficult to measure but that does not lessen that productive value of well managed welfare systems. Another example of stress is that experienced by pigs that are susceptible to heat stress due to lack of sweat glands. They will wallow in mud to lower their body temperature when under heat stress and it is claimed (Curtis & Stricklin, 1991) that they will not wallow even if they have access to mud if the temperature is optimal for them at 12 degrees Celsius. He also acknowledged evidence that pigs reared in an enriched more natural environment were more productive.

The argument about animals having human emotions (Tangley, 2000), from cats purring to grieving elephants,(Starr, 2006) or chimpanzees (Bekoff, 2000) following the death of their offspring, is a powerful indication of the strength of emotional attachments of animals from birth. It has been demonstrated that attachment can be interfered with in animals the same as with humans as Harry Harlow found (Van, LeRoy, & Van, 2008), however it can also be used in a positive way by utilizing natural imprinting processes (Manning, 1967) to achieve training outcomes. It could also be claimed that broken attachments can have permanent effects on a juvenile animal.

Harlow found beyond question that by disrupting the mother infant bond, the infants did not learn to mother their own offspring adequately and while some ignored their own babies at birth, others killed them. John Bowlby, (van & van, 2010) observed the effects of disrupted attachments in children at the clinic he worked at and it was undeniable in the face of the evidence he presented, that human children, the same as animal babies, suffered from permanent psychological disorders such as anger, despair and separation anxiety as a result (Steele, 2010). He was also interested in Konrad Lorenz's work on imprinting goslings and realized that in animals as in humans, (Sable, 2004) that bonding is part of a basic survival instinct since infants are nurtured by parents when ill, threatened by danger or predator attack and the close proximity of a parent means safety can be assured. Lorenz, who together with Niko Tinbergen  discovered how the phenomenon of imprinting in animals naturally occurs.

As stated, physiological adaptations enable animals to survive in the wild and to find food, water and flee from danger. They have highly equipped senses which are processed in the brain through the  nervous system. The sensory system includes the receptor cells and sense organs, the  central nervous system which is the brain and the spinal cord and the motor system which controls the motor systems such as the autonomic and somatic systems. To understand animal emotion, a little must be understood about the processes of the brain and the differences between prey animal and predator physiology and psychology.

The sensory input of a prey animal is different and their perception is faster than a predator, as they must be able to quickly flee from danger or it will get eaten, (Miller, 1999). They have a highly developed sense of smell,  hearing and  sight and the visual system of a prey animal is designed differently to a predator.

The majority of prey animals have eyes on the side of their head and horses in particular, have the largest eyes of any terrestrial mammal. They have almost 350 degrees of vision giving them excellent panoramic and distance vision, (McGreevy, 2004). They have mostly monocular vision (McGreevy, 2004) and they have a special layer in the eye called the “tapetum lucidum” which is designed to reflect light onto the retina making it easy for horses to see in dim light (Mills & Nankervis, 1999). The hearing of a horse is far more superior than that of a human, in the range of 1-16 kHz which is a broader range than most mammals, making them extremely responsive to strange noises that could indicate danger (McGreevy, 2004). They also have an extremely sensitive olfactory system enabling stallions to distinguish when mares are in estrus at breeding time as do predator species.

Predators such as birds of prey have eyes on the front of their head giving them excellent depth perception and  binocular vision with excellent  visual acuity, enabling them to quickly judge exactly where their prey is in order to catch it. They also have a highly developed sense of smell enabling them also to detect food, such as seen in some breeds of dogs used for tracking and to detect and mark territory. Their hearing is also highly developed and many animal species have hearing far superior to humans.

Emotions seen in animals are evident in both prey and predators species (Carmichael, Reno, & Shenfeld, 2003). Dogs are extremely adept at reading human emotions as claimed by Carmichael (2003) in her article about animal emotions, she also claims that this adaptation has enabled dogs to find food and shelter with their human companions in early caveman days. She posited that dogs have personalities, sociability, affection, emotional stability and competence, or a combination of obedience and intelligence. [] The video link is of a dog and other animals who lost their babies and adopted others. The second video link is of a tiger who adopted piglets, [] as the story goes, the tiger lost her own cubs and became depressed according to the zoo keepers, so they introduced piglets who needed care and she accepted them as her own, also recovered from her depression, which is compelling evidence that animals feel emotions similar to humans!

The amygdala in the brain is said to be responsible for the emotion of love or intense affection. Carmichael, (2003) recalled incidents of monkeys adopting babies unrelated to them, whales doing courtship dances and horses refusing to leave their stalls due to an intense bond they have formed. While animals do form bonds with humans, it is questionable that the reason the horses don’t wish to leave their stalls is because of the bond, it could also be other underlying motivators keeping them in there, such as fear.

Fear is claimed to be an innate characteristic that animals have in order to survive, they do not need to think about danger when it is their instinct to react first in order to escape, (Bekoff, 2000). An example of this built in alarm system would be certain sounds, objects flying around or smells. In the limbic system the amygdala is the part of the brain responsible for responding to a stimulus and causing an emotional response such as flight from fear, which is the physiological response, as it is a primary defence mechanism in prey species. For example, if a zebra was to think first about a noise in the bush and run later, then it would most likely get eaten, but if it runs first then thinks later, it increases its chances of survival. This sort of adaptation can be utilized by trainers who are aware of the psychology behind the animal’s reactions to elicit responses. An example would be in the zebra’s domestic cousin, the horse. Through using certain techniques such as imprinting young, habituation and desensitization (Miller, 2007), amazing results can be achieved.

The fear response can also turn into anger or extremely aggressive behaviour, like a beaten dog for example. It is afraid of its owner, so when the owner approaches the dog, it perceives a threat because it is cornered and reacts with anger or aggression because it has no other escape. Examples of animal anger are demonstrated in this video clip where enraged circus elephants run amok, [] where animals attack their owners, supposedly as a result of the harsh training methods used. Other videos linked to in this chapter show unprovoked attacks,[] however, those animals were not necessarily beaten, there is no suggestion they were, but there was a reason they attacked their owners.

In considering animal welfare, the question of why some animals display stereotypical behaviour such as self mutilation, weaving or pacing the fence need to be addressed as they are definitely caused by emotional stress of some description. [](The video is as it is approx 6 min, but the first example gives a good demonstration. Measuring emotions in animals is difficult but warrant further scientific analysis to reduce these behaviours in captive animals from a welfare perspective.

Bringing man and animals together - interspecies communication
In relation to inter-species communication from a motivation and emotion perspective, understanding the psychology of prey animals and predators and how they perceive humans is of great relevance. A prey animal knows humans are predators, therefore they can be expected to be fearful of humans the majority of the time. They may not run away but they may not exactly trust the human either. If they are imprinted at birth this can greatly reduce their fear and make them more trainable.

In the supplementary section there are examples of training methods that are based on building communication across species. Clicker training, classical and operant conditioning, appropriate reinforcement techniques that work with the animal's natural instincts rather than against them are discussed. Animals understand body language and have personalities, they are also extremely intelligent. They are motivated by physiological needs such as hunger, thirst, sex, companionship, air and rest. They are also motivated to escape from danger, but can be taught to become less afraid and do things normally unnatural to them (such as go on a horse float). They can be so well taught using the right motivational techniques that they will actively seek out the float as it can provide comfort - comfort, (to stop and rest) is a huge incentive.

Conclusion
Motivation and emotion in animals is inextricably linked and one directly affects the other. It has been shown that animals can think, demonstrating cognition, they also have personalities. There is very strong evidence to support the notion that they experience emotions similar to humans. Their emotional state under stress can be psychological as well as physiological. Some of the psychological effects of stress manifest themselves as stereotypical behavior. Animals are motivated by the need for survival, those drives are extremely powerful, or example, a prey animal will run until it needs to stop for air, then that will become more important. Sometimes they will fight if they see no other way out. Understanding emotional states in animals, drive theory, learning theory, imprinting techniques and to consider personalities and how they perceive humans is paramount to understanding what motivates them.

Utilising appropriate shaping and reinforcement techniques, in combination with the appopriate classical or operant training (or a combination of the two), using flooding, approach retreat and habituation techniques will almost certainly cure the majority of animal behaviour issues people face. Applying the concepts outlined in the supplementary section that are used on horses could be applied to any prey animal.

People who work with animals in zoos, animal parks and beef/cattle/dairy industries would benefit greatly from this knowledge as it would benefit the comfort and welfare of animals overall. Animals like rhinoceros and elephants could be imprinted at birth, followed up with teaching them to lift their feet on cue or move wherever the human wants. This training would make them less fearful and better prepared for general handling such as that required when the vet visits. It would also make them safer to handle. Even reptiles such as crocodiles can be trained using positive reinforcement, this can be used to the handler's advantage. Animals can be taught anything from tricks, circus work and movie work to be world class competition animals.

Clicker training is an excellent training method for some predatory animals such as killer whales and dolphins. Operant/classical conditioning used appropriately using negative and positive reinforcement has been demonstrated as highly effective on animals such as big cats. Embracing training during critical early learning periods can pay dividends later on, for both predator and prey animals. Understanding the use of body language and understanding animal's body language is a nonverbal form of interspecies communication. Animals can be communicated to by body language and other subtle cues, such as with circus animals and movie / liberty trained animals.

Hopefully this chapter encourages people to explore further and keep an open mind when exploring "new" modern methods of training. Understanding the predator / prey animal relationship and individual differences amongst animals is the most important message. Any animal lover / owner should know these concepts, animals do not need to be bullied or brutalised to submit to the will of humans. The methods described here are far more humane and consider the world from the animal's point of view while using positive reinforcement at the appropriate time. Humans keep animals in unnatural environments and expect them to cope with it, the least we can do is prepare them psychologically, it is our responsibility.

Animals can be dangerous and there is always a risk of injury and no liability will be entered into from any individual attempting methods mentioned here as this chapter is not intended to replace professional guidance, only provide general information. Readers are encouraged seek professional instruction.


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User talk:Cgonzal4

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User talk:Jasmine13

Human Motivation

-What is motivation? (Johnson, 1997) - "Motivation as an inferred need, desire or impulse which initiates, directs and sustains behaviour" - a factor which explains the relations between stimuli and behaviour

-Self Determination Theory (SDT) Deci, E.L., & Ryan, R.M. (2008). Self-determination theory: A macrotheory of human motivation, development and health). Canadian Psycology, 49(3), 182-185.

 * Differentiates types of motivation, particularly focusing on autonomous and controlled motivation, and amotivation as predictors of performance and well-being outcomes.
 * Focus' not on varying strengths of needs in individuals, but rather the degree to which the need has been satisfied in the individual.
 * 2 general individual differences concepts: 1)causality orientations, and 2) life goals

-Intrinsic & extrinsic motivation

Biological Aspect of Sexual Motivation (Wallen, 2002) "Gonadal hormones primarily influence sexual motivation" as they regulate a mammals ability to copulate. Eliminating testosteronein adult humans and rhesus monkeys, decreases stimulation and therefore, sexual motivation. Human sexual behaviour is influenced by 3 main factors: hormonal state, social context, and cultural conventions. A female's sexual desire is influenced dramatically by ovarian hormones, however, it is suggested that cognition also plays a significant role in human sexual behaviour as the percieved pregnancy risk also influences specific sexual behaviours that individuals participate in.

(Sherwin, Gelfand, & Brender) Studies by these guys with menopausal women, and women who had undergone a hysterectomy, showed a clear increase in the intensity of sexual arousal and desire in participants due to exogenous androgen. However, testosterone did not appear to affect physiological responses or interpersonal aspects of sexual behaviour. The conclusion made by these researchers, therefore, was that androgen acted significantly on sexual motivation rather than sexual activity.

Sexual Motivation in Humans (Johnson, 1997) In lower animals, sexual motivation is seen as a 'drive' or, an internal force that "pushes" the animal to "engage in reproductive behaviour". Human sexual motivation is a much more complicated process involving several complex factors. Building on the definition of motivation by Johnson (1997) and applying it to human motivation, it can be said to be "an inferred, internal state influenced by several factors which determines engagement in sexual activity"

Methods in data collection for human sexual motivation & desire Most research is done via surveys, self-reports and volunteers due to the private, personal nature of the topic. However, this leaves room for many errors either via socially-desired responses, unintentional errors or miscommunication or misunderstanding about what an individual may think motivates their behaviour. Also, often participants in these studies are seen as 'ungeneralisable' to the population as they are often much more outgoing about sex and related topics. Most studies are correlational, not causal as they cannot "control for all potential variables"

2 Landmark studies Kinsey studies - "highly influencial due to the nature of the questions asked and the large number (several thousand) of subjects who were polled". Aimed to "identify, among other facts, what sexual behaviours people engaged in, what age they were when they began engaging in them, and how often they were currently engaging in them" Found physiology of both sexes to be quite similar, differing mainly in women's lesser sexual capacity. Caused uproar in the conservative times (1950s). Kinsey studies disregarded effects of culture and learning on the results of their subject's behaviours.

Masters & Johnson - involved "actual physiological measurement of sexual responses in male and female volunteers" Proved the similarity between both sexes and their sexual response to intercourse, contradictory to what was originally thought. "Their results would indicate that differential subjective responses to sexual intercourse between the sexes were indeed more likely associated with culture and learning"

Whilst now the data from both studies is considered outdated and not generalisable (white, middle class volunteers), provided an important backdrop for recent research and studies.

One in United States and one in America - used non-volunteers and extensive interviews. "These studies were designed to include a representative sample and also allow participants to give in-depth, and anonymous answers. Laumann and his associates found a more conservative pattern of sexual behaviour than did the Kinsey studies indicating that volunteers are not, in fact, representative of the general population."

Gender differences in motivation for engaging in sexual activity.
 * Males
 * Females
 * Homosexual/Heterosexual. Is there any difference?

Motivation is a factor which explains the relations between stimuli and behaviour; motivation is inferred need, desire or impulse which initiates, directs and sustains behaviour (Coon, 1997). “Human motivation to engage in sexual behaviour is due to a complex relationship among several factors” Klusmann, Dietrich. (2002). Sexual motivation and the duration of partnership. Archives of Sexual Behaviour, 31(3), 275-287. •	Surveyed 1865 German students aged 19-32 who reported to be heterosexual and in a steady relationship. Based on cross-sectional data meaning a longitudinal explanation is unsafe •	Main results: 1.	“sexual activity and sexual satisfaction decline in women and men as the duration of the partnership increases” 2.	“sexual desire only declines in women” 3.	“desire for tenderness declines in men and rises in women” •	Klusmann suggest the reason for such findings was due to individual differences in mating strategies •	Suggested evolutionary explanation for results: 1.	The attachment and bond felt between individuals in a partnership developed from a non-sexual parent-child tie. 2.	Therefore, once initial infatuation period has passed between a couple, their relationship is seen to become less sexual. 3.	However, evolutionary history suggests that a male’s sexual desire should stay at a higher level due to sperm/reproductive competition 4.	In comparison, female sexual desire is “assumed to reflect an adaptive function – to boost attachment and secure the bond” •	As the term of the relationship increased, sexual motivation for women decreased, but not for men. •	This paper looks at 3 main factors in order to explain results: 1.	Results from other surveys 2.	Social psychology 3.	Theories about evolutionary origins of gender differences in sexual motivation •	“Sexual behaviour often requires a compromise between the different proclivities of women and men (Symons, 1979), but sexual desire and other emotions, which are antecedent or contingent to sexual behaviour are not restricted in this way” •	“Thus, gender differences are expected mainly to appear in the frequency of emotions, not behaviours” •	HOW DOES EMOTION AND SATISFACTION CONTRIBUTE TO MOTIVATION? NB: This article looks into emotion! •	Look at EVOLUTIONARY THEORY •	Sexual desire vs sexual motivation??? •	Does different types of sexual activity contribute to motivation to engage?? •	Active & passive stimulation •	“The decline of sexual desire in women was strongly revealed in the assessment of sexual desire relative to the partner and in an avowed lack of sexual desire... few existing differences point to less female motivation and pleasure in longer relationships, in comparison to shorter ones” •	“Partners in a couple often differ in their commitment. One of them may exceed the other in 1.	Wishing the partner to endure 2.	Wanting to possess the partner for herself or himself 3.	Wanting to spend much time with the partner 4.	Wanting to talk to the partner 5.	Being jealous, and 6.	Being in love •	Stated negatively, 1.	Fear of intimacy may be elicited more easily in one of the partners 2.	He or she may tend to withdraw emotionally” •	“imbalance of commitment related to sexual desire...” 1.

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User:Egm6321.f10.team1.allen

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title: User:Egm6321.f10.team1.allen


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User talk:Egm6321.f10.team1.allen

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User:Egm6321.fall10.team1.lang

HW1

title: User:Egm6321.fall10.team1.lang


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User:Egm6321.f10.team1.lang/HW1

To prove that $${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}$$ is a nonlinear equation with respect to $${{y}^{1}}$$, we will consider a specific case. In an effort to make obvious the nonlinear nature of the of the previous equation we choose the trigonometric function sine to be included in our definition of $${{c}_{3}}({{y}^{1}},t)$$.

Let

$${{c}_{3}}(y,t)=\sin (y)+{{t}^{2}}$$

Next we need to define a criterion to determine whether or not a given equation is linear or nonlinear. For L to be considered a linear operator, it must have the following properties:

$$L(\phi +\psi )=L(\phi )+L(\psi )$$

$$L(k\phi )=kL(\phi )$$

where k is any scalar.

In order to test our newly formed rules for linearity (and simplify the writing), lets express the equation

$${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}=[\sin ({{y}^{1}})+{{t}^{2}}]\frac{d{{t}^{2}}}$$

as an operator L.

Let

$$L(\cdot )=[\sin (\cdot )+{{t}^{2}}]\frac{{{d}^{2}}(\cdot )}{d{{t}^{2}}}$$

We may now apply the prior definitions for linearity as a test for whether L is linear or nonlinear. First we check if the distributive property holds by computing the LHS of the defining equation for linear operators.

$$L(\phi +\psi )=[\sin (\phi +\psi )+{{t}^{2}}]\frac{{{d}^{2}}(\phi +\psi )}{d{{t}^{2}}}$$

and using the trigonometric identity for sine:

$$\sin (\phi +\psi )=\sin (\phi )\cos (\psi )+\cos (\phi )\sin (\psi )$$

we arrive at the following:

$$=[\sin (\phi )\cos (\psi )+\cos (\phi )\sin (\psi )+{{t}^{2}}][\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]$$

which then becomes

$$=[\sin (\phi )\cos (\psi )[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]+\cos (\phi )\sin (\psi )[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]+{{t}^{2}}[\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\frac{{{d}^{2}}\psi }{d{{t}^{2}}}]$$

and expanding into the 6 term expression we have

$$=[\sin (\phi )\cos (\psi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\sin (\phi )\cos (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+\cos (\phi )\sin (\psi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\cos (\phi )\sin (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

We will see in a momment why I picked the trigonometric funcation sine to demonstrate a test for linearity of an operator. Now we compute the RHS of the defining equation for linear operators.

$$L(\phi )+L(\psi )=[\sin (\phi )+{{t}^{2}}]\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+[\sin (\psi )+{{t}^{2}}]\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

distributing the derivatives over the sums we arrive at the following 4 term expression

$$=\sin (\phi )\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\phi }{d{{t}^{2}}}+\sin (\psi )\frac{{{d}^{2}}\psi }{d{{t}^{2}}}+{{t}^{2}}\frac{{{d}^{2}}\psi }{d{{t}^{2}}}$$

As we can see, the nonlinearities in L with respect to $${{y}^{1}}$$ are obvious in the sine term, and L as defined does not distribute over the given sum. If L did distribute, then we would have seen equivalent expressions between both the RHS and LHS of the defining equation for linearity (i.e. the RHS gave 6 terms which were not equivalent to the 4 terms the LHS gave). Therefore, $${{c}_{3}}({{y}^{1}},t)\frac{d{{t}^{2}}}$$ is a nonlinear equation with respect to $${{y}^{1}}$$.

Egm6321.fall10.team1.lang 07:22, 14 September 2010 (UTC)

title: User:Egm6321.f10.team1.lang/HW1


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User talk:Katelin Barker90

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User:Moofirst

WAT PHRA SRI RATTANA MAHATHAT

History

The temple was built in the time of Phra Maha Dhamma Raja (King Li Thai) alias Phra Chao Sri Dhamma Tri Pidok when the town of Songkwai(Phitsanulok) was built in 1357 (B.E.1900). The Northern history stated that. This temple is called by general people as " WAT YAI" or "WAT LUANG POR PHRA BUDDHA CHINNARAJ" Phrabat Somdej Phra Mongkutklao ( King Rama VI ) classified as " Vora Maha Viharn" or the first grade of Royal Temple in the year of 1915 (B.E.2458). At the present, Wat Yai is named " Wat Phra Sri Rattana Mahathat" Wat Phra Sri or Wat Yai situates on the east bank of Nan River opposite the Phitsanulok City Hall. Prince Damrong Rajanupab said about Wat Phra Sri Rattana Mahathat. In the year of 1357 Phra Chao Sri Dhamma Tri Pidok (Phra Maha Dhamma Raja ,King Li Thai) was a pious king, a true believer in the Buddhism. He studied the Tri-Pidok and teachings of other religions fluently. He built up Wat Phra Sri Mahathat on the east side with a Pagoda in the middle compounds, 4 Viharns in the 4 corners with 2 terraces and casted 3 Buddha images as Main Buddha in the temple. There are many antiques and valuable of unestimated value in the temple. Somdej Phra Chao Boromwong Ther Krom Phraya. "After paying homage to Phra Buddha Chinnaraj, I took notice of pulpits, the pavilion and others, I found them to be of an exquisitely designs I have never met before. "The temple is a big and more important than other temples in Phitsanulok and is believed that was built in the Sukhothai period being renovated form time to time".

Wat Phra Sri Rattana Mahathat is an important Temple, not only to the people of Phitsanulok, but also to Buddhists throughout Thailand. Every King of the Chakri dynasty including the present King, H.M.King Bhumiphol Adulyadej has come to this Temple to worship in front of the Phra Budda Chinnarat image. The Temple is also an important tourist attraction and at times can be extremely busy with many visitors both Thai and non-Thai.

title: User:Moofirst


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User talk:Zero-one

 Hello Zero-one, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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Template:X1/doc

Experimental template documentation.

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