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User:Egm6321.f09.team3.diggs/HW6

= Circular Cylindrical Coordinates =

Given
$$x_1 = x = r \cos \theta = \xi_1 \cos \xi_2$$

$$x_2 = y = r \sin \theta = \xi_1 \sin \xi_2$$

$$x_3 = z = z = \xi_3$$

Find
a) $${dx_i}$$ in terms of $${\xi_j}$$ and $${d \xi_k}$$

b) Find $$ds^2 = \sum (dx_i)^2 = \sum(h_k)^2(d \xi)^2$$. Identify $${h_i}$$ in terms of $${\xi_i}$$.

c) Find the Laplacian in cylindrical coordinates

d) Obtain separated equations for the Laplace Equation in circular cylindrical coordinates and identify the Bessel differential equation of the form $$x^2 y'' + x y - (x^2 - \nu^2) = 0$$.

Solution
a)

$$dx_1 = d( \xi_1 \cos \xi_2) = d\xi_1 \cdot \cos \xi_2 $$

$$dx_2 = d( \xi_1 \sin \xi_2) = d \xi_2 \cdot \xi_1 \cdot \cos \xi_2$$

$$dx_3 = d(\xi_3) = d\xi_3$$

$${dx_i} = (\cos \xi_2 \cdot d\xi_1, \xi_1 \cdot \cos \xi_2 \cdot d\xi_2, d\xi_3)$$

b)

$$ds^2 = \sum(dx_1)^2 = (\cos \xi_2)^2 (d\xi_1)^2 + (\xi_1)^2(\cos \xi_2)^2 (d\xi_2)^2 + (1)^2 \cdot (d\xi_3)^2$$

where $${h} = (\cos \xi_2, \xi_1 \cdot \cos \xi_2, 1)$$

c)

$$ \Delta \Psi = \frac{1}{h_1 \cdot h_2 \cdot h_3} \sum_{i=1}^3 \frac {\partial}{\partial \xi_i} \left[ \frac{h_1 \cdot h_2 \cdot h_3}{(h_i)^2} \cdot \frac {\partial \psi}{\partial \xi_i}\right] $$

where $$h_1h_2h_3 = \xi_1 \cdot (\cos \xi_2)^2$$

$$ \Delta \Psi = \frac{1}{\xi_1 \cdot (\cos \xi_2)^2} \left[ \frac {\partial}{\partial \xi_1} \left[ \frac{\xi_1 \cdot (\cos \xi_2)^2}{(\cos \xi_2)^2} \cdot \frac {\partial \psi}{\partial \xi_1}\right] + \frac {\partial}{\partial \xi_2} \left[ \frac{\xi_1 \cdot (\cos \xi_2)^2}{(\xi_1 \cdot \cos \xi_2)^2} \cdot \frac {\partial \psi}{\partial \xi_2}\right] + \frac {\partial}{\partial \xi_3} \left[ \frac{\xi_1 \cdot (\cos \xi_2)^2}{(1)^2} \cdot \frac {\partial \psi}{\partial \xi_3}\right] \right] $$

$$ \Delta \Psi = \frac{1}{\xi_1 \cdot (\cos \xi_2)^2} \left[ \frac {\partial}{\partial \xi_1} \left( \xi_1 \frac {\partial \psi}{\partial \xi_1}\right) + \frac{1}{\xi_1} \frac {\partial^2 \psi}{\partial \xi_2^2} + \xi_1 \cos^2 \xi_2 \frac {\partial^2 \psi}{\partial \xi_3^2} \right]$$

$$ \Delta \Psi = \frac{1}{\xi_1 \cdot \left(\cos \xi_2\right)^2} \frac {\partial}{\partial \xi_1} \left( \xi_1 \frac {\partial \psi}{\partial \xi_1}\right) + \frac{1}{\xi_1^2 \cdot (\cos \xi_2)^2} \frac {\partial^2 \psi}{\partial \xi_2^2} + \frac {\partial^2 \psi}{\partial \xi_3^2} $$

d)

Assume axisymmetric, not a function of $$ \xi_3 $$

$$ \Delta \Psi = \frac{1}{\xi_1 \cdot (\cos \xi_2)^2} \frac {\partial}{\partial \xi_1} ( \xi_1 \frac {\partial \psi}{\partial \xi_1}) + \frac{1}{\xi_1^2 \cdot (\cos \xi_2)^2} \frac {\partial^2 \psi}{\partial \xi_2^2} $$

Let $$psi(\xi_1, \xi_2) = \psi(r, \theta) = R(r) \Theta(\theta)$$

$$ \Delta \Psi = \frac{\Theta}{r \cdot (\cos \theta)^2} \frac {d}{dr} ( r \frac {dR}{dr}) + \frac{R}{r^2 \cdot (\cos \theta)^2} \frac {d^2 \Theta}{d\theta^2} = 0 $$

Multiply by $$ r^2 (\cos \theta)^2 $$

$$ 0 = r \Theta \frac{d}{dr} (r \frac{dR}{dr}) + R \frac{d^2 \Theta}{d \theta^2}$$

Divide by $$R\Theta$$

$$ 0 = \frac{r}{R} \frac{d}{dr} (r \frac{dR}{dr}) + \frac{1}{\Theta} \frac{d^2 \Theta}{d \theta^2}$$

Now the equation is of the form $$0 = \alpha(r) + \beta(\theta)$$ which yields $$\alpha(r) = -\beta(\theta) = k$$, where k is constant

the function in terms of r is:

$$\frac{r}{R} \frac{d}{dr} (r \frac{dR}{dr}) = k$$

$$r \frac{d}{dr} (r \frac{dR}{dr}) = kR$$

$$r(r R'' + R') - kR = 0$$

$$r^2R'' + rR' - kR = 0$$

where $$k = (x^2 - \nu^2)$$ to get the form of the Bessel equation

for completeness, the function in terms of $$\theta$$ is:

$$\frac{1}{\Theta} \frac{d^2\Theta}{d\theta^2} = -k$$

$$\Theta'' + k\Theta = 0$$

= Spherical Coordinates in Astronomy Notation=

Given
$$x_1 = x = r \cos \theta \cos \varphi= \xi_1 \cos \xi_3 \cos \xi_2$$

$$x_2 = y = r \cos \theta \sin \varphi = \xi_1 \cos \xi_3 \sin \xi_2$$

$$x_3 = z = r \sin \theta = \xi_1 \sin \xi_3$$

Find
a) $${dx_i}$$ in terms of $${\xi_j}$$ and $${d \xi_k}$$

b) Find $$ds^2 = \sum (dx_i)^2 = \sum(h_k)^2(d \xi)^2$$. Identify $${h_i}$$ in terms of $${\xi_i}$$.

c) Find the Laplacian and compare to the solution in King

Solution
a)

$$dx_1 = d( \xi_1 \cos \xi_2 \cos \xi_3) = d\xi_1 \cdot \cos \xi_2 \cos \xi_3 $$

$$dx_2 = d( \xi_1 \sin \xi_2 \cos \xi_3) = d \xi_2 \cdot \xi_1 \cdot \cos \xi_2 \cos \xi_3$$

$$dx_3 = d(\xi_1 \sin \xi_3) = d\xi_3 \xi_1 \cos \xi_3$$

$${dx_i} = (\cos \xi_2 \cos \xi_3 \cdot d\xi_1, \xi_1 \cdot \cos \xi_2 \cdot \cos \xi_3 \cdot d\xi_2, \xi_1 \cos \xi_3 d\xi_3)$$

b)

$$ds^2 = \sum(dx_1)^2 = (\cos \xi_2)^2 (\cos \xi_3)^2 (d\xi_1)^2 + (\xi_1)^2(\cos \xi_2)^2 (\cos \xi_3)^2 (d\xi_2)^2 + (\xi_1)^2 (\cos \xi_3)^2 \cdot (d\xi_3)^2$$

where $${h} = (\cos \xi_2 \cdot \cos \xi_3, \xi_1 \cdot \cos \xi_2 \cdot \cos \xi_3, \xi_1 \cdot \cos \xi_3)$$

c)

$$ \Delta \Psi = \frac{1}{h_1 \cdot h_2 \cdot h_3} \sum_{i=1}^3 \frac {\partial}{\partial \xi_i} \left[ \frac{h_1 \cdot h_2 \cdot h_3}{(h_i)^2} \cdot \frac {\partial \psi}{\partial \xi_i}\right] $$

where $$h_1h_2h_3 = \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^3$$

$$ \Delta \Psi = \frac{1}{\xi_1^2 \cdot (\cos \xi_2)^2 cdot (\cos \xi_3)^3} \left[ \frac {\partial}{\partial \xi_1} \left[ \frac{\xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^3} { (\cos \xi_2 \cdot \cos \xi_3)^2 } \cdot \frac {\partial \psi}{\partial \xi_1}\right] + \frac {\partial}{\partial \xi_2} \left[ \frac{\xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^3}{( \xi_1 \cdot \cos \xi_2 \cdot \cos \xi_3 )^2} \cdot \frac {\partial \psi}{\partial \xi_2}\right] + \frac {\partial}{\partial \xi_3} \left[ \frac{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^3 }{( xi_1 \cdot \cos \xi_3 )^2} \cdot \frac {\partial \psi}{\partial \xi_3}\right] \right] $$

$$ \Delta \Psi = \frac{1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^3 } \left[ \cos \xi_3 \frac {\partial}{\partial \xi_1} ( \xi_1^2 \frac {\partial \psi}{\partial \xi_1} + \cos \xi_3 \frac {\partial^2 \psi}{\partial \xi_2^2} + (\cos \xi_2)^2 \frac {\partial}{\partial \xi_3} ( \cos \xi_3 {\partial \psi}{\partial \xi_3} \right]$$

$$ \Delta \Psi = \frac{1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^2 } \frac {\partial}{\partial \xi_1} ( \xi_1^2 \frac {\partial \psi}{\partial \xi_1} + \frac{1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\cos \xi_3)^2 }  \frac {\partial^2 \psi}{\partial \xi_2^2} + \frac{1}{ \xi_1^2 \cdot (\cos \xi_3)^2 }  \frac {\partial}{\partial \xi_3} ( \cos \xi_3 {\partial \psi}{\partial \xi_3} $$

Converting back to $$(r, \varphi, \theta)$$

$$ \Delta \Psi = \frac{1}{r^2 \cos^2 \varphi \cos^2 \theta} \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \cos^2 \varphi \cos^2 \theta} \frac{\partial ^2 \psi}{\partial \varphi^2} + \frac{1}{r^2 \cos^3 \theta} \frac{\partial}{\partial \theta} (\cos \theta \frac{\partial \psi}{\partial \theta}) $$

I have the same thing as you up to here. I am not sure that by assuming that is axisymetrical you can just eliminate the $\cos^2 \varphi$ on the first term. I know that the second term goes to zero but not sure about the first term. And I am not matching the formula that the instructor gave us maybe i am not understanding something here?

Assume axisymmetric (not a function of $$\varphi$$)

$$ \Delta \Psi = \frac{1}{r^2 \cos^2 \theta} \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \cos^3 \theta} \frac{\partial}{\partial \theta} (\cos \theta \frac{\partial \psi}{\partial \theta}) = 0 $$

Multiply by $$ \cos^2 \theta $$

$$ \frac{1}{r^2 } \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \cos \theta} \frac{\partial}{\partial \theta} (\cos \theta \frac{\partial \psi}{\partial \theta}) = 0 $$

Compare to King, page 45,

$$ \frac{1}{r^2 } \frac{\partial}{\partial r}(r^2 \frac{\partial T}{\partial r}) + \frac{1}{r^2 \sin \bar \theta} \frac{\partial}{\partial \bar \theta} (\sin \bar \theta \frac{\partial T}{\partial \bar \theta}) = 0 $$

\since $$\sin \bar \theta = \sin(\frac {\pi}{2} - \theta) = \cos \theta$$, the equations match!

= Spherical Coordinates in Math/Physics Notation =

Given
$$x_1 = x = r \sin \bar\theta \cos \varphi= \xi_1 \sin \xi_3 \cos \ xi_2$$

$$x_2 = y = r \sin \bar\theta \sin \varphi = \xi_1 \sin \xi_3 \sin \xi_2$$

$$x_3 = z = r \cos \bar \theta = \xi_1 \cos \xi_3$$

Find
a) $${dx_i}$$ in terms of $${\xi_j}$$ and $${d \xi_k}$$

b) Find $$ds^2 = \sum (dx_i)^2 = sum(h_k)^2(d \xi)^2$$. Identify $${h_i}$$ in terms of $${\xi_i}$$.

c) Find the Laplacian and compare to the solution in King

Solution
a)

$$dx_1 = d( \xi_1 \cos \xi_2 \sin \xi_3) = d\xi_1 \cdot \cos \xi_2 \sin \xi_3 $$

$$dx_2 = d( \xi_1 \sin \xi_2 \sin \xi_3) = d \xi_2 \cdot \xi_1 \cdot \cos \xi_2 \sin \xi_3$$

$$dx_3 = d(\xi_1 \cos \xi_3) = d\xi_3 \xi_1 (- \sin \xi_3)$$

$${dx_i} = (\cos \xi_2 \sin \xi_3 \cdot d\xi_1, \xi_1 \cdot \cos \xi_2 \cdot \sin \xi_3 \cdot d\xi_2, \xi_1 (-\sin \xi_3) d\xi_3)$$

b)

$$ds^2 = \sum(dx_1)^2 = (\cos \xi_2)^2 (\sin \xi_3)^2 (d\xi_1)^2 + (\xi_1)^2(\cos \xi_2)^2 (\sin \xi_3)^2 (d\xi_2)^2 + (\xi_1)^2 (-\sin \xi_3)^2 \cdot (d\xi_3)^2$$

where $${h} = (\cos \xi_2 \cdot \sin \xi_3, \xi_1 \cdot \cos \xi_2 \cdot \sin \xi_3, - xi_1 \cdot \sin \xi_3)$$

c)

$$ \Delta \Psi = \frac{1}{h_1 \cdot h_2 \cdot h_3} \sum_{i=1}^3 \frac {\partial}{\partial \xi_i} \left[ \frac{h_1 \cdot h_2 \cdot h_3}{(h_i)^2} \cdot \frac {\partial \psi}{\partial \xi_i}\right] $$

where $$h_1h_2h_3 = - \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^3$$

$$ \Delta \Psi = \frac{1}{- \xi_1^2 \cdot (\cos \xi_2)^2 cdot (\sin \xi_3)^3} \left[ \frac {\partial}{\partial \xi_1} \left[ \frac{-\xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^3} { (\cos \xi_2 \cdot \sin \xi_3)^2 } \cdot \frac {\partial \psi}{\partial \xi_1}\right] + \frac {\partial}{\partial \xi_2} \left[ \frac{-\xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^3}{( \xi_1 \cdot \cos \xi_2 \cdot \sin \xi_3 )^2} \cdot \frac {\partial \psi}{\partial \xi_2}\right] + \frac {\partial}{\partial \xi_3} \left[ \frac{ -\xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^3 }{( xi_1 \cdot \sin \xi_3 )^2} \cdot \frac {\partial \psi}{\partial \xi_3}\right] \right] $$

$$ \Delta \Psi = \frac{-1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^3 } \left[ -\sin \xi_3 \frac {\partial}{\partial \xi_1} ( \xi_1^2 \frac {\partial \psi}{\partial \xi_1} + -\sin \xi_3 \frac {\partial^2 \psi}{\partial \xi_2^2} + -(\cos \xi_2)^2 \frac {\partial}{\partial \xi_3} ( \sin \xi_3 {\partial \psi}{\partial \xi_3} \right]$$

$$ \Delta \Psi = \frac{1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^2 } \frac {\partial}{\partial \xi_1} ( \xi_1^2 \frac {\partial \psi}{\partial \xi_1} + \frac{1}{ \xi_1^2 \cdot (\cos \xi_2)^2 \cdot (\sin \xi_3)^2 }  \frac {\partial^2 \psi}{\partial \xi_2^2} + \frac{1}{ \xi_1^2 \cdot (\sin \xi_3)^3 }  \frac {\partial}{\partial \xi_3} ( \sin \xi_3 {\partial \psi}{\partial \xi_3} $$

Converting back to $$(r, \varphi, \theta)$$

$$ \Delta \Psi = \frac{1}{r^2 \cos^2 \varphi \sin^2 \bar\theta} \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \cos^2 \varphi \sin^2 \bar \theta} \frac{\partial ^2 \psi}{\partial \varphi^2} + \frac{1}{r^2 \sin^3 \bar \theta} \frac{\partial}{\partial \bar\theta} (\sin \bar \theta \frac{\partial \psi}{\partial \bar \theta}) $$

Assume axisymmetric (not a function of $$\varphi$$)

$$ \Delta \Psi = \frac{1}{r^2 \sin^2 \bar \theta} \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \sin^3 \bar \theta} \frac{\partial}{\partial \bar \theta} (\sin \bar \theta \frac{\partial \psi}{\partial \bar \theta}) = 0 $$

Multiply by $$ \sin^2 \bar \theta $$

$$ \frac{1}{r^2 } \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) + \frac{1}{r^2 \sin \bar \theta} \frac{\partial}{\partial \bar \theta} (\sin \bar \theta \frac{\partial \psi}{\partial \bar \theta}) = 0 $$

Compare to King, page 45,

$$ \frac{1}{r^2 } \frac{\partial}{\partial r}(r^2 \frac{\partial T}{\partial r}) + \frac{1}{r^2 \sin \bar \theta} \frac{\partial}{\partial \bar \theta} (\sin \bar \theta \frac{\partial T}{\partial \bar \theta}) = 0 $$

The equations match!

= Solution by Quadratic Equation =

Given
$$\lambda (\lambda +1 ) = n(n +1)$$

Find
Solutions $$\lambda = n$$ and $$\lambda = -(n + 1)$$

Solution
Using the equation $$\frac{-b +/- \sqrt{4ac - b^2}}{2a}$$

where a = 1, b = 1, c = -n(n+1),

$$\lambda = \frac{-b +/- \sqrt{4ac - b^2}}{2a} = \frac{-1 +/- \sqrt{4(1)(-n(n+1) - 1}}{2 \cdot 1}$$

$$\lambda = \frac{-b +/- \sqrt{4ac - b^2}}{2a} = \frac{-1 +/- (2n + 1)}{2}$$

u\sing the + solution:

$$\lambda = \frac{-1 + (2n + 1)}{2} = n$$

u\sing the - solution:

$$\lambda = \frac{-1 - (2n + 1)}{2} = - (n + 1)$$

= Prove Polynominal Statements Agree =

Given
$${P}_{0} = 1$$

$${P}_{1} = x$$

$${P}_{2} = \frac{1}{2}(3{x}^{2} - 1)$$

$${P}_{3} = \frac{1}{2}(5{x}^{2} - 3x)$$

$${P}_{4} = \frac{35}{8}{x}^{4} - \frac{15}{4}{x}^{2} + \frac{3}{8}$$

And,

$$ {P}_{n}(x) = \sum_{i=0}^{[\frac{n}{2}]}{(-1)}^{i}\frac{(2n-2i)!*{x}^{n-2i}}{{2}^{n}*i!(n-i)!(n-2i)!}$$

Show
The polynomials $$ {P}_{0}, {P}_{1}, {P}_{2}, {P}_{3}, {P}_{4}$$ are in agreement with the summation in $${P}_{n}$$.

Solution
''I have worked the problem from both directions and this is the best way I could get agreement, feel free to edit, correct, or discard. I have finished most of the assignment on paper and will be uploading my solutions (for what is still left blank) over the next 2 days. I am done for today though. - Barrett ''

$$ {P}_{n}(x) = \sum_{i=0}^{[\frac{n}{2}]}{(-1)}^{i}\frac{(2n-2i)!*{x}^{n-2i}}{{2}^{n}*i!(n-i)!(n-2i)!}$$

For n=0,

$$ {P}_{0}(x) = \sum_{i=0}^{[\frac{0}{2}]}{(-1)}^{0}\frac{(2*0-2*0)!*{x}^{0-2*0}}{{2}^{0}*0!(0-0)!(0-2*0)!}$$

(this is solved knowing 0! = 1)

$$ {P}_{0}(x) = 1*\frac{1!*1}{1!*1!*1!} = 1 $$, this is same as eq(1) pg 31-3 from the notes.

For n = 1,

$$ {P}_{1}(x) = {(-1)}^{0}\frac{(2*1-2*0)!*{x}^{1-2*0}}{{2}^{1}*0!(1-2*0)!(1-2*0)!}$$

$$ {P}_{1}(x) = 1*\frac{2*x}{2} = x $$

For n = 2,

$$ {P}_{2}(x) = \sum_{i=0}^{1}{(-1)}^{i}\frac{(2*2-2i)!*{x}^{2-2i}}{{2}^{2}*i!(2-i)!(2-2i)!}$$

$$ {P}_{2}(x) = {(-1)}^{0}\frac{(2*2-2*0)!*{x}^{2-2*0}}{{2}^{2}*0!(2-0)!(2-2*0)!} + {(-1)}^{1}\frac{(2*2-2*1)!*{x}^{2-2*1}}{{2}^{2}*1!(2-1)!(2-2*1)!}$$

$$ {P}_{2}(x) = \frac{24}{16}*{x}^{2}+\frac{1}{2} = \frac{1}{2}(3*{x}^{2} + 1)$$

For n = 3,

$$ {P}_{3}(x) = \sum_{i=0}^{1}{(-1)}^{i}\frac{(2*3-2i)!*{x}^{3-2i}}{{2}^{3}*i!(3-i)!(3-2i)!}$$

$$ {P}_{3}(x) = {(-1)}^{0}\frac{(2*3-2*0)!*{x}^{3-2*0}}{{2}^{3}*0!(3-0)!(3-2*0)!} + {(-1)}^{1}\frac{(2*3-2*1)!*{x}^{3-2*1}}{{2}^{3}*1!(3-1)!(3-2*1)!}$$

$$ {P}_{3}(x) = \frac{720}{288}*{x}^{3}+\frac{3}{2}*x = \frac{1}{2}(5*{x}^{2} + 3*x)$$

For n = 4,

$$ {P}_{4}(x) = \sum_{i=0}^{2}{(-1)}^{i}\frac{(2*4-2i)!*{x}^{4-2i}}{{2}^{4}*i!(4-i)!(4-2i)!}$$

$$ {P}_{4}(x) = {(-1)}^{0}\frac{(2*4-2*0)!*{x}^{4-2*0}}{{2}^{4}*0!(4-0)!(4-2*0)!} + {(-1)}^{1}\frac{(2*4-2*1)!*{x}^{4-2*1}}{{2}^{4}*1!(4-1)!(4-2*1)!} + {(-1)}^{2}\frac{(2*4-2*2)!*{x}^{4-2*2}}{{2}^{4}*2!(4-2)!(4-2*2)!}$$

$$ {P}_{4}(x) = \frac{35}{8}*{x}^{4} - \frac{15}{4}*{x}^{2} + \frac{3}{8} $$

= Verify Polynominal Statement is a Solution of the Legendre Equation =

Given
The Legendre Equation

$$ (1-x^2)y''-2xy'+n(n+1)y=0$$

and the Legendre Polymomial statement

$${P}_{0} = 1$$

$${P}_{1} = x$$

$${P}_{2} = \frac{1}{2}(3{x}^{2} - 1)$$

$${P}_{3} = \frac{1}{2}(5{x}^{2} - 3x)$$

$${P}_{4} = \frac{35}{8}{x}^{4} - \frac{15}{4}{x}^{2} + \frac{3}{8}$$

Find
The solution such that the polynominal statement agrees

Solution
$$

P_0=1$$

$$P_0'=P_0''=0 $$

Substituting P0 for y yields

$$ (1-x^2)(0)-2x(0)+0(0+1)1=0$$

so P0 satisfies the Legendre Equation

$$ P_1=x $$

$$P_1'=1$$

$$P_1''=0 $$

Substituting P1 for y yields

$$ (1-x^2)(0)-2x(1)+1(1+1)x=0$$

$$ -2x+2x=0$$

so P1 satisfies the Legendre Equation

$$ P_2=\frac{1}{2}(3x^2-1)$$

$$P_2'=3x$$

$$P_2''=3 $$

Substituting P2 for y yields

$$ (1-x^2)(3)-2x(3x)+2(2+1)(\frac{1}{2}3x^2-1)=0$$

$$ 3-3x^2-6x^2+9x^2-3=0$$

so P2 satisfies the Legendre Equation

$$ P_3=\frac{1}{2}(5x^3-3x)$$

$$P_3'=\frac{15}{2}x^2-\frac{3}{2}$$

$$P_3''=15x $$

Substituting P3 for y yields

$$ (1-x^2)(15x)-2x(\frac{15/2}x^2-\frac{3}{2} )+3(3+1)(\frac{1}{2}(5x^3-3x) =0$$

$$ 15x-15x^3-15x^3+3x+30x^3-18x=0$$

so P3 satisfies the Legendre Equation

$$ P_4=\frac{35}{8}x^4-\frac{15}{4}x^2+\frac{3}{8} $$

$$P_4'=\frac{35}{2}x^3-\frac{15}{2}x$$

$$P_4''=\frac{105}{2}x^2-\frac{15}{2} $$

Substituting P4 for y yields

$$ (1-x^2)(\frac{105}{2}x^2-\frac{15}{2} )-2x(\frac{35}{2}x^3-\frac{15}{2}x )+4(4+1)(\frac{35}{8}x^4-\frac{15}{4}x^2+\frac{3}{8}) =0$$

$$ \frac{105}{2}x^2-\frac{105}{2}x^4-\frac{15}{2} +\frac{15}{2}x^2-\frac{70}{2}x^4+\frac{30}{2}x^2+\frac{175}{2}x^4-\frac{150}{2}x^2+\frac{15}{2}=0$$

so P4 satisfies the Legendre Equation

= Odd/Even Functions =

Given
A function is even if $$ f(-x)= f(x)$$

A function is odd if $$ f(-x) = -f(x)$$

Find
Show that the sum of 2 odd functions is odd

Solution
Let $$f = \sum g_i$$

If $${g_1}$$ is odd, then f is odd.

If $${g_i}$$ is even, then f is even.

= Proof of Polynominal Statement as Odd/Even =

Given
$$P_{2k}$$ and $$P_{2k+1}$$

Find
$$P_{2k}$$ is even

$$P_{2k+1}$$ is odd

Solution
The polynominal statement is:

$$ {P}_{n}(x) = \sum_{i=0}^{[\frac{n}{2}]}{(-1)}^{i}\frac{(2n-2i)! \cdot {x}^{n-2i}}{{2}^{n} \cdot i!(n-i)!(n-2i)!}$$

For n = 2k, the polynomial statement becomes:

$$ {P}_{2k}(x) = \sum_{i=0}^{[\frac{2k}{2}]}{(-1)}^{i}\frac{(2 \cdot 2k -2i)! \cdot {x}^{2k-2i}}{{2}^{2k} \cdot i!(2k-i)!(2k-2i)!}$$

Because the definition of odd or even depends upon only the function of x, we can rename the portion of $$ {P}_{2k}(x) $$ that depends on x as:

$$ p_{2k}(x) = \sum_{i=0}^{[k]}x^{2k-2i}$$

for any value of k, $$p_{2k}$$ includes an x to an even power, which makes the whole function, $${P}_{2k}$$, even!

For n = 2k+1, the polynomial statement becomes:

$$ {P}_{2k+1}(x) = \sum_{i=0}^{[\frac{2k+1}{2}]}{(-1)}^{i}\frac{(2 \cdot (2k+1) -2i)! \cdot {x}^{2k + 1 -2i}}{{2}^{2k+1} \cdot i!(2k+1-i)!(2k+1-2i)!}$$

Because the definition of odd or even depends upon only the function of x, we can rename the portion of $$ {P}_{2k}(x) $$ that depends on x as:

$$ p_{2k+1}(x) = \sum_{i=0}^{[\frac {2k+1}{2}]}x^{2k+1-2i}$$

for any value of k, $$p_{2k+1}$$ includes an x to an odd power, which makes the whole function, $${P}_{2k+1}$$, odd!

= Find Coefficients =

Given
$$q(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4 x^4$$

and

$$c_0 = 3 $$

$$c_1 = 10$$

$$c_2 = 15$$

$$c_3 = -1$$

$$c_4 = 5$$

Find
Find $$a_i$$ such that $$q = \sum a_i P_i$$.

Plot $$q = \sum c_i x^i = \sum a_i P_i$$.

Solution
Since $$q(x) = 3 + 10 x + 15 x^2 - x^3 + 5x^4$$

and $$ q = a_0 \cdot P_0 + a_1 \cdot P_1 + a_2 \cdot P_2 + a_3 \cdot P_3 + a_4 \cdot P_4 $$

Note $$P_0, P_1, P_2, P_3, P_4, P_4$$ have been solved in prior homework problems.

Solve using matrices, such that:

$$\begin{bmatrix} 3\\ 10x \\ 15x^2 \\ -x^3 \\ 5x^4 \end{bmatrix} = \begin{bmatrix} a_0\\ a_1 \\ a_2 \\ a_3\\ a_4 \end{bmatrix} \begin{bmatrix} 1 & 0 & -\frac{1}{2} & 0 & \frac{3}{8} \\ 0 & x & 0 & \frac{3}{2} x^2 & 0 & -\frac{15}{4}x^2 \\ 0 & 0 & 0 & \frac{5}{2} x^3 & 0 \\ 0 & 0 & 0 & 0 & \frac{35}{8}x^4 \end{bmatrix}$$

Solving for the column matrix of a values:

$$\begin{bmatrix} a_0\\ a_1 \\ a_2 \\ a_3\\ a_4 \end{bmatrix} = \begin{bmatrix} 9 \\ 9.4 \\ 12.857 \\ -0.4 \\ 1.1428 \end{bmatrix}$$

This yields two equations for q:

$$q = \sum c_i x^i = 3 + 10x + 15x^2 - x^3 + 5x^4$$

$$q = 9(1) + 9.4 x + 12.857 (\frac{1}{2}(3x^2 -1)) - 0.4(\frac{1}{2}(5x^2 - 3x)) + 1.1428(\frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8})$$

Plot the two solutions to ensure they match:

= HW 4 Question, King 1.1.b =

Given
$$u_1(x) = \frac{1}{x}\sin x$$

and the ODE

$$xy'' + 2y' + xy = 0$$

Find
Prove $$u_1(x) = \frac{1}{x}\sin x$$

and then find $$u_2(x)$$ u\sing the method of reduction of order 2

Solution
First, determine if the ODE is exact:

where $$ f(x,y,p) = x $$ and $$ g(x,y,p) = 2p+xy $$

Condition 2.i is:

$$ f_{xx} + 2p \cdot f_{xy} + p^2 \cdot f_{yy} = g_{xp} + p \cdot g_{yp} - g_y $$

and the partial derivatives are:

$$ f_{xx} = f_{xy} = f_{yy} = g_{xp} = g_{yp} = 0 $$

or $$ g_y = x $$

which yields

$$ 0 + 2p \cdot 0 + p^2 \cdot 0 = 0 + p \cdot 0 - x $$

which is not satified, therefore there's no reason to go on to the second condition.

Next, we can find the integrating factor ($$h(x,y,) = x^my^n$$) to make the equation exact OR use the method of trial solution.

(I tried the integration factors method and could not find a value of m to satisfy, so we'll move directly into the method of trial solutions.)

In the method of trial solution, first try $$ y(x) = exp(rx) $$

Take the first and second derivatives and substitute back into the ODE, cancelling the term $$ exp(rx) $$ to get:

$$ xr^2 + 2r + x = 0 $$

\since the solution requires r = f(x), we move on to another trial solution.

Our second trial solution is $$ y(x) = x \cdot exp(rx) $$

Take the first and second derivatives and substitute back into the ODE, cancelling the term $$ exp(rx) $$ to get:

$$ x^2r^2 + 4rx + x^2 + 2 = 0 $$

which again requires r = f(x), so we move on to a third trial solution.

Our third trial solution is $$ y(x) = \frac{1}{x} \cdot exp(rx) $$

Take the first and second derivatives and substitute back into the ODE, cancelling the term $$ exp(rx) $$ to get:

$$ r^2 + 1 = 0 $$

which is $$ r = +/- i $$

By deMoivre, the solution is

$$u_1(x) = \frac{1}{x} \sin x$$

Now, \since we have $$u_1(x)$$, we need to find $$u_2(x)$$ u\sing the reduction of order method 2

We assume the full solution $$ y = U(x) u_1(x) $$

\since the process is the same as shown in Problem #1, we go straight to the solution:

$$ u_2(x) = u_1(x) \int \frac{1}{u_1(t)^2} exp(- \int a_1(s) ds) dt $$

where $$ a_1(s) = \frac{2}{s}$$

such that $$ u_2(x) = \frac{1}{x} \sin x \int \frac{1}{(\sin t)^2} dt = -\frac{\cos(x)}{x} $$

Therefore, $$ u_2(x) =-\frac{\cos(x)}{x} $$

= Properties and Calculation of A(n) =

Given
$$\psi(1, \theta) = f(\theta) = T_0 \cos \theta$$

or

$$\psi(1, \theta) = f(\mu) = T_0 \sqrt{1- \mu^2}$$

and

$${A}_{n} = \frac{2n+1}{2}\int_{-1}^{1}f(\mu){P}_{n}(\mu) d\mu$$

Find
a) without calculation, the properties of $$A_n$$, i.e., $$A_{2k} = 0$$? or $$A_{2k+1} = 0$$?

b) Compute 3 non-zero coefficients of $$A_n$$, using (i) analytical methods (e.g. integration table) and (ii) numerically using Gauss_Legendre, accurate to 5%

Solution
a) $$ f(\mu) = {T}_{o}{(1-{\mu}^{2})}^{\frac{1}{2}}$$ is even and is an element of a set of Polynomials of degree 2 or less. Therefore it can be concluded $$ {A}_{n} = 0 $$ for all $$ n = 2k+1 $$ where the polynomial expressions are odd. Also, it is only necessary to evaluate $${A}_{0}$$ and $${A}_{2}$$.

b) (i) Using analytical integration techniques the two even (non-zero) coefficients are solved as follows:

 * $${A}_{0} = \frac{1}{2}*\int_{-1}^{1}({T}_{0}{(1-{\mu}^{2})}^{\frac{1}{2}}{P}_{0}(\mu)) d\mu$$, where $${P}_{0}(\mu) = 1$$

using the integration tables,

 * $${A}_{0} = \frac{1}{2}*\frac{\pi{T}_{0}}{2} = \frac{\pi{T}_{0}}{4} $$

for the next non-zero coefficient,

 * $${A}_{2} = \frac{5}{2}\int_{-1}^{1}({T}_{0}{(1-{\mu}^{2})}^{\frac{1}{2}}{P}_{2}(\mu)) d\mu$$, where $${P}_{2}(\mu) = \frac{1}{2}(3{\mu}^{2} - 1)$$

using the integration tables,

 * $${A}_{2} = \frac{5}{4}*\frac{-\pi{T}_{0}}{8} = \frac{-5\pi{T}_{0}}{32} $$

 * $${A}_{4} = \frac{9}{2}*\int_{-1}^{1}({T}_{0}{(1-{\mu}^{2})}^{\frac{1}{2}}{P}_{4}(\mu)) d\mu$$, where $${P}_{4}(\mu) = \frac{35}{8}{\mu}^{4} - \frac{15}{4}{\mu}^{2} + \frac{3}{8}$$

using the integration tables,

 * $${A}_{4} = \frac{9}{2}*\frac{-\pi{T}_{0}}{128} = \frac{-9\pi{T}_{0}}{256} $$

(ii) Using Gauss-Legendre the two even (non-zero) coefficients are computed numerically as follows:

 * $${A}_{n} = \frac{2n+1}{2}\int_{-1}^{1}f(\mu){P}_{n}(\mu) d\mu = \frac{2n+1}{2}\sum_{i=1}^{n}{w}_{i}f({x}_{i}) $$

 * $$ w_i = \frac{2}{\left( 1-x_i^2 \right) (P'_n(x_i))^2} \,\!$$

for n = 2,

 * $$A_2 = \frac{2(2)+1}{2}\sum_{i=1}^{2}(\frac{2}{\left( 1-x_i^2 \right) (P'_2(x_i))^2}f(x_i))$$

 * $$A_2 = \frac{5}{2}*2*T_0\sum_{i=1}^{2}\frac{(1-{x_i}^{2})^{\frac{1}{2}}}{(1-{x_i}^{2})*9{x_i}^{2}} $$

$$x_i$$ is the roots of the Legendre polynomial, therefore $$x_i = \pm\sqrt{1/3}$$

 * $$A_2 = 5T_0(.408248 - .288675) = .597866T_0 $$

for n = 4,

 * $$A_4 = \frac{2(4)+1}{2}\sum_{i=1}^{4}(\frac{2}{\left( 1-x_i^2 \right) (P'_4(x_i))^2}f(x_i))$$

 * $$A_4 = \frac{9}{2}*2*T_0\sum_{i=1}^{2}\frac{(1-{x_i}^{2})^{\frac{1}{2}}}{(1-{x_i}^{2})*{(\frac{35}{2}x_i^3 - \frac{15}{2}x_i)}^{2}} $$

$$x_i$$ is the roots of the Legendre polynomial, therefore $$x_i = \pm\sqrt{\Big( 3 \pm 2\sqrt{6/5} \Big)/7}$$

= Gauss- Legendre Quadrature =

Given
The Legendre Polynomial

Find
a) Verify the table in Wikipedia (analytical expressions for $$x_j$$, $$w_j$$ for j = 1, ..., n; n = 1, ..., 5).

b) Verify the expressions for $$P_n$$ with n=0, 1, ..., 6

c) Evaluate numerically $$x_j$$, $$w_j$$ and compare results with Abram and Stegun

Solution
The Wikipedia table in question is:

For n=1, $$P_1(x) = x$$, let $$P_1(x) = x = 0$$ to find the roots, such that the root is $$x_1 = 0$$.

$$P_1'(x) = 1$$

$$P_2(x) = \frac{1}{2} (3x^2 - 1)$$

$$w_1(x=0) = \frac{-2}{(2)(1)(\frac{1}{2}(3(0)-1))} = \frac{-2}{-1} = 2$$

as expected from the table.

For n = 2, $$P_2(x) = \frac{1}{2} (3x^2 - 1)$$, let $$P_2(x) = \frac{1}{2} (3x^2 - 1) = 0$$ to find the roots, such that the roots are $$x_{1,2} = +/- \frac{1}{\sqrt {3}}$$.

$$P_2'(x) = 3x$$

$$P_3(x) = \frac{1}{2} (5x^3 - 3x)$$

$$w_{1,2}(x=+/- \frac{1}{\sqrt {3}}) = \frac{-2}{(3)(3(+/- \frac{1}{\sqrt {3}}))(\frac{1}{2}(5(+/- \frac{1}{\sqrt {3}})^3-3(+/- \frac{1}{\sqrt {3}})))} = 1$$

as expected from the table.

For n = 3, $$P_3(x) = \frac{1}{2} (5x^3 - 3x)$$, let $$P_3(x) = \frac{1}{2} (5x^3 - 3x) = 0$$ to find the roots, such that the roots are $$x_{1,2} = +/- \sqrt{\frac{3}{5}}$$, $$x_3 = 0$$.

$$P_3'(x) = \frac{15}{2}x^2 - \frac{3}{2}$$

$$P_4(x) = \frac{35}{8}x^4 - \frac{15}{4}x^2 - \frac{3}{8} $$

$$w_{1,2}(x=+/- \sqrt{\frac{3}{5}}) = \frac{-2}{(4)(7.5(+/- \sqrt )- \frac{3}{2})(\frac{1}{8}(35(+/- \sqrt{\frac{3}{5}})^4-30(+/- \sqrt{\frac{3}{5}}) - 3))} = \frac{10}{63}$$

$$w_3(x=0) = \frac{-2}{(4)(7.5(0)- \frac{3}{2})(\frac{1}{8}(35(0)^4-30(0) - 3))} = -\frac{8}{9} $$

WHICH DOES NOT MATCH the table.

Need n = 4

Need n = 5

b)

Show $$P_n$$ follows the table for n = 0, ..., 6

$$ {P}_{n}(x) = \sum_{i=0}^{[\frac{n}{2}]}{(-1)}^{i}\frac{(2n-2i)!*{x}^{n-2i}}{{2}^{n}*i!(n-i)!(n-2i)!}$$

For n=0,

$$ {P}_{0}(x) = \sum_{i=0}^{[\frac{0}{2}]}{(-1)}^{0}\frac{(2*0-2*0)!*{x}^{0-2*0}}{{2}^{0}*0!(0-0)!(0-2*0)!}$$

(this is solved knowing 0! = 1)

$$ {P}_{0}(x) = 1*\frac{1!*1}{1!*1!*1!} = 1 $$, this is same as eq(1) pg 31-3 from the notes.

For n = 1,

$$ {P}_{1}(x) = {(-1)}^{0}\frac{(2*1-2*0)!*{x}^{1-2*0}}{{2}^{1}*0!(1-2*0)!(1-2*0)!}$$

$$ {P}_{1}(x) = 1*\frac{2*x}{2} = x $$

For n = 2,

$$ {P}_{2}(x) = \sum_{i=0}^{1}{(-1)}^{i}\frac{(2*2-2i)!*{x}^{2-2i}}{{2}^{2}*i!(2-i)!(2-2i)!}$$

$$ {P}_{2}(x) = {(-1)}^{0}\frac{(2*2-2*0)!*{x}^{2-2*0}}{{2}^{2}*0!(2-0)!(2-2*0)!} + {(-1)}^{1}\frac{(2*2-2*1)!*{x}^{2-2*1}}{{2}^{2}*1!(2-1)!(2-2*1)!}$$

$$ {P}_{2}(x) = \frac{24}{16}*{x}^{2}+\frac{1}{2} = \frac{1}{2}(3*{x}^{2} + 1)$$

For n = 3,

$$ {P}_{3}(x) = \sum_{i=0}^{1}{(-1)}^{i}\frac{(2*3-2i)!*{x}^{3-2i}}{{2}^{3}*i!(3-i)!(3-2i)!}$$

$$ {P}_{3}(x) = {(-1)}^{0}\frac{(2*3-2*0)!*{x}^{3-2*0}}{{2}^{3}*0!(3-0)!(3-2*0)!} + {(-1)}^{1}\frac{(2*3-2*1)!*{x}^{3-2*1}}{{2}^{3}*1!(3-1)!(3-2*1)!}$$

$$ {P}_{3}(x) = \frac{720}{288}*{x}^{3}+\frac{3}{2}*x = \frac{1}{2}(5*{x}^{2} + 3*x)$$

For n = 4,

$$ {P}_{4}(x) = \sum_{i=0}^{2}{(-1)}^{i}\frac{(2*4-2i)!*{x}^{4-2i}}{{2}^{4}*i!(4-i)!(4-2i)!}$$

$$ {P}_{4}(x) = {(-1)}^{0}\frac{(2*4-2*0)!*{x}^{4-2*0}}{{2}^{4}*0!(4-0)!(4-2*0)!} + {(-1)}^{1}\frac{(2*4-2*1)!*{x}^{4-2*1}}{{2}^{4}*1!(4-1)!(4-2*1)!} + {(-1)}^{2}\frac{(2*4-2*2)!*{x}^{4-2*2}}{{2}^{4}*2!(4-2)!(4-2*2)!}$$

$$ {P}_{4}(x) = \frac{35}{8}*{x}^{4} - \frac{15}{4}*{x}^{2} + \frac{3}{8} $$

For n = 5,

$$ {P}_{5}(x) = \sum_{i=0}^{2}{(-1)}^{i}\frac{(2*5-2i)!*{x}^{5-2i}}{{2}^{5}*i!(5-i)!(5-2i)!}$$

$$ {P}_{5}(x) = {(-1)}^{0}\frac{(2*5-2*0)!*{x}^{5-2*0}}{{2}^{5}*0!(5-0)!(5-2*0)!} + {(-1)}^{1}\frac{(2*5-2*1)!*{x}^{5-2*1}}{{2}^{5}*1!(5-1)!(5-2*1)!} + {(-1)}^{2}\frac{(2*5-2*2)!*{x}^{5-2*2}}{{2}^{5}*2!(5-2)!(5-2*2)!}$$

$$ {P}_{5}(x) = \frac{1}{8}*(63{x}^{5} - 70*{x}^{3} + 15x) $$

For n = 6,

$$ {P}_{6}(x) = \sum_{i=0}^{3}{(-1)}^{i}\frac{(2*6-2i)!*{x}^{6-2i}}{{2}^{6}*i!(6-i)!(6-2i)!}$$

$$ {P}_{6}(x) = {(-1)}^{0}\frac{(2*6-2*0)!*{x}^{6-2*0}}{{2}^{6}*0!(6-0)!(6-2*0)!} + {(-1)}^{1}\frac{(2*6-2*1)!*{x}^{6-2*1}}{{2}^{6}*1!(6-1)!(6-2*1)!} + {(-1)}^{2}\frac{(2*6-2*2)!*{x}^{6-2*2}}{{2}^{6}*2!(6-2)!(6-2*2)!}  + {(-1)}^{3}\frac{(2*6-2*3)!*{x}^{6-2*3}}{{2}^{6}*3!(6-3)!(6-2*3)!}$$

$$ {P}_{6}(x) = \frac{1}{16}*(231{x}^{6} - 315*{x}^{4} + 105x^2 - 5) $$

= Proof of Inverse Hyperbolic Tangent =

Given
$$tanh^{-1}(x)=\left[\frac{1}{2}log\left(\frac{1+x}{1-x}\right)\right]$$

Show
Derivation of the equation

Solution
$$tanh s = \frac{e^s-e^{-s}}{e^s+e^{-s}} $$

let $$s=log t$$

$$ tanh (log t) = \frac{e^{(log t)} -e^{-(log t) }}{e^{(log t)} +e^{-(log t) }}$$

$$ tanh (log t) = \frac{t-\frac{1}{t}}{t +\frac{1}{t}} =\frac{t^2-1}{t^2+1}$$

It follows that

$$ tanh \left [\frac{1}{2}log\left(\frac{1+x}{1-x}\right)\right]=tanh\left(log\sqrt{\frac{1+x}{1-x}}\right)=\frac{\frac{1+x}{1-x}-1}{\frac{1+x}{1-x}+1}= \frac{2x}{2}=x$$

Taking the inverse:

$$tanh^{-1}(x)=\left[\frac{1}{2}log\left(\frac{1+x}{1-x}\right)\right]$$

title: User:Egm6321.f09.team3.diggs/HW6


text:
User:Wikiresources/Introduction

Introduction
Wikiresources is a Wikiversity page created with the objective to explore, evaluate and record knowledge about diverse resources applying to elearning.

This assignment is part of one of the requirements and learning outcomes of the MSc. in Applied eLearning, in the Dublin Institute of Technology, Dublin, Ireland. In the Instructional Design and eAuthoring module, the students will be expected to work as a group to design and develop an activity/elearning resource that can be used as a tool for learning.

Our group choose the wiki software Wikiversity because it seems to be the most appropriate way to record and disseminate the knowledge acquired during this Master Degree course.

The intention of this wiki page is to provide a structure within which students place their knowledge, ideas, experiences and conclusions about the different resources available to elearning. Therefore this is not a conclusive work as it will be continuously updated by users. From this structure the students can include or change concepts, images, videos or any other learning object according their studies and reflections. Consequently it could be used as a resource for learning for future students of the course.

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

title: User:Wikiresources/Introduction


text:
Template:Make a portal/Tab4

Usage
See the instructions at Make a portal.

title: Template:Make a portal/Tab4


text:
Template:Make a portal/Tab5

Usage
See the instructions at Make a portal.

title: Template:Make a portal/Tab5


text:
User:Babalon~enwikiversity

Time's glory is to command contending kings, To unmask falsehood, and bring truth to light.

title: User:Babalon~enwikiversity


text:
User:Meryle

I'm a web designer from upstate New York, interested in online learning.

title: User:Meryle


text:
User:CRBR

About Me    Contributions to Wikiversity

About Me

 * 1) Junior in High School
 * 2) Want to severely contribute to Electrical Engineering and any COmputer Science Projects

title: User:CRBR


text: User talk:Clblogmail

title: User talk:Clblogmail


text:
User:Clblogmail

祈祷中文维基学院早日脱离 Beta，我在英语维基学院等着你！

http://zh.wikiversity.org/，目前还是Wiki does not exist！

title: User:Clblogmail


text:
User talk:Harryc181

 Hello Harryc181, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 11 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 11 November 2009 (UTC)

title: User talk:Harryc181


text:
User talk:Incran888

 Hello Incran888, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 11 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 11 November 2009 (UTC)

title: User talk:Incran888


text:
User talk:Landscape~enwikiversity

 Hello Landscape, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 11 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 11 November 2009 (UTC)

Your account will be renamed
Hello,

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

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

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

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:36, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:14, 19 April 2015 (UTC)

title: User talk:Landscape~enwikiversity


text:
Learning management systems vs. social network systems

Please edit and improve - this is a wiki

James Neill's response
I am interested to see a couple of edtech heavyweights (Gsiemens and Martin Dougiamas slug it out on this gsiemens blog post about learning management systems versus social network systems - but sometimes you'd think these guys have never heard of a wiki?

@gsiemens and @martindougiamas - thanks for this discussion. My main concern with both LMS and SNS approaches is the lack of primacy given to collaborative editably of learning content - in the LMS (such as Moodle) what's possible is rather centralised and generally locked down - a "teacher" can't do much except add a few links to boxes and embed some stuff. A "learner" can do even less. In a social network, there's connections and conversations everywhere - in fact this seems to be what is priveleged - but (at least with CCK09) the whole things seems a bit "floaty" - what about getting down to collaboratively creating and editing shared content? I'm wondering if maybe connectivism is significantly North American? (Individualism + Conversation - penchant for talk therapy etc.) At least I found in my experiences of USA educational style, that talking and discussion is very important to the educational culture. But, after all the talk, the action seems problematic. Also, a difficulty I've had with CCK09 is that "readings" are seemingly locked down to editing (but can be converted/forked by the intrepid) and there is limited (practical) availability of recordings etc. (in a limited range of closed formats).

I enjoy your edtech work, guys, and the thinking and discussion that goes with it, but I puzzle and scratch my head about how come we don't just learn together in real-world open spaces where anyone can edit (and without using proprietary software) forever? Maybe we're grappling like blind men with the elephant - I reckon you guys have hold of pieces of the elephant - and you're right. And I hope its the elephants nuts I've got hold of and not yours. In blindness. -- Jtneill - Talk - c 08:43, 11 November 2009 (UTC)

George Siemens's response
Hi James - I had to chuckle at your final statement. Took a bit of time to sink in. ouch. Hope it is the elephant.

The biggest problem that I see with networked learning right now is what we are experiencing now - fragmented discussions that are not aware of each other. The mix (and the magic) is in the aggregation. Aggregation can create a similar sense of space to what a centralized forum has now. But, participating in networks of distributed content/discussions is a big cognitive shift for most of people.

I don't think connectivism is unique to North America. We constantly connect and exchange information. Whether or not we indulge in additional discussion/debate may be more NA & Europe centric. But the core of connecting remains.

Good point about readings being locked down. That has to do more with our need to meet U of Manitoba's criteria for a course syllabus than our own views. The limited range of formats of recordings - again, excellent point. We rely on Elluminate for recording the sessions. They have a file conversion tool...but it take time to run. Last year we encouraged attendees to make their own copies of sessions. The limit here is again not ideological - it's technological.

Beyond SNS-views of learning, I agree that greater collaborative activity needs to be promoted and fostered. Wikis are a great example of this process. Regrettably, most wiki software is not well understood by the majority of educators (either conceptually or technically - wiki language can be a bit challenging for some). But, as we more forward with next generation learning software, we cannot overlook the importance of collaboration...


text: File:Funnel questions.png

title: File:Funnel questions.png


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University of Canberra/LEAG

Learning Environments Advisory Group
This group seeks to facilitate the creation and nurturing of conducive on-campus and online learning environments at the University of Canberra.

title: University of Canberra/LEAG


text:
Talk:Technical writing/Requirements and Functional Specifications EE

Snoozebob 12:37, 11 November 2009 (UTC)


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Template:Atp/theory

The content is provided without any warranties of fitness for a particular purpose or suitability for pupils in a specific grade.

Parameters:
 * The license parameter can be set to "gfdl+cc" to indicate that a specific resource is not released into the public domain but dual-licensed under the GNU Free Documentation license and the Creative Commons license. If the parameter is not given the license will default to public domain. The parameter can also be set to "none" to indicate that no license information should be displayed.

title: Template:Atp/theory


text: Category:Theory Design Lab

title: Category:Theory Design Lab


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Talk:Tourism/Stages

Where would you say Lisbon is on the Butler Model? —The preceding unsigned comment was added by 92.14.210.230 (talk • contribs) 16:51, 11 November 2009 (UTC)

title: Talk:Tourism/Stages


text:
Template:Atp/subject

 This resource is intended for use in the subject . This template is used for categorising subject specific resources of the Theory Design Lab.

title: Template:Atp/subject


text:
User:DanielleFulgham

=Freshman Seminar=

Always Wanted To Schedule

 * Topic

To learn how to play Whale and Wasp by Alice in Chains on an acustic guitar.
 * Research Component

The history of the band Alice and Chains, as well as other bands during the time they started. Also more information about guitar playing styles.
 * week 1

Look up Alice and Chains Whale and Wasp song, try to find music sheet.
 * week 2

Practice the different notes and fingerings, do a bit of research on AiC.
 * week 3

Listen to the song several times to better understand the tempo, research on styles of playing.
 * week 4

Practice more, research other similar bands.
 * week 5

Figure out how to record myself playing, and set-up presentation.
 * week 6

Final practice, and recording.
 * week 7

Set-up presentation on laptop, possible power point.
 * week 8

Final touches, practice presentation.

Midterm Assignment 1: Taste & Aesthetics
The sculpture I dislike at the Seattle Sculpture park was ‘Typwriter Eraser, Scale X’ and one of the artists that worked on it was Claes Oldenburg. I’ve seen this sculpture many times, driving or walking past it. It seems so odd to me that it’s pretty much in the middle of nowhere, and it makes no sense to me. I don’t feel anything when I look at it, and when I look at art I expect to have some sort of emotion exasperated. The movement in the piece is decent but I just don’t understand it’s purpose.

When I looked up more of his work, I found a piece called ‘Soft Bathtub’. Now this one strikes me as something very interesting. It looks like a real bathtub from afar, but it’s made with canvas, wood and plaster. I’m not sure why but it does gives me an eerie feeling. It reminds of the Silent Hill movie with the way it’s distorted and how dirty it looks. I’ve always been a fan of horror films and creepy looking things so naturally it catches my eye. I love that piece by Oldenburg, but I just can’t bring myself to like the ‘Typerwriter Eraser‘. I’ll admit he is a creative man, but I still think the piece is completely random and meaningless. But maybe that’s the purpose of it.

Midterm Assignment 2: Manipulation
http://www.youtube.com/watch?v=-ijH3cRJ1BY

http://runeechan.deviantart.com/art/Rant-and-Rave-10704161

Midterm Assignment 3: Always Wanted To Progress Report (about 100 wds)
I had troubles getting the Alice in Chains Whale and Whasp music sheets. I tried to find free versions online but only found interpretations and strange codes from students. And also I have been procrastinating a little bit on getting the music for it. I don't have the money to go out and buy a music book, so I think I'll just work on a song I had learned back in High School. "House of the Rising Sun" by The Animals. I will research that band instead and practice the song. I already have the music sheet for it, so I won't have to worry much about that aspect of it. Just the practicing and research.

Final Presentation

 * Topic (10%)

"House of the Rising Sun" by The Animals

 * Research (20%)

<>

 * 1) Internet site:

"The Animals: Biography : Rolling Stone." Rolling Stone: Music News, Reviews, Photos, Videos, Interviews, Politics and More. Web. 02 Dec. 2009. .
 * 2) Book: Animals, The. Hutchinson Unabridged Encyclopedia. Helicon Publishing. 2005. eLibrary. ProQuest LLC. ART INSTITUTE OF SEATTLE. 02 Dec 2009. .
 * 3) Scholarly Journal: Jansen, Lloyd. "Don't Let Me Be Misunderstood." Library Journal. 2001. 126. eLibrary. ProQuest LLC. ART INSTITUTE OF SEATTLE. 02 Dec 2009. .
 * 4) Other (interview, TV show,etc):"The Animals." Rock and Roll Hall of Fame and Museum. Web. 02 Dec. 2009. .

 * Your accomplishment (20%)

I practiced the song many times, and got the notes down, although I did procrastinate a lot due to World of Warcraft. What I need to master is the fingering for the song instead of just strumming it. I used to know how to do that when I took the guitar class in High School, but have forgotten it since. My previous project idea what the Alice in Chains song Whale and Whasp, which I would still like to learn how to play once I can acquire the music for it.

 * Project Review (50%)
 * 1)What went well: Playing the song went better than I had hoped. I haven't taken a class for guitar in a long time, and have been a bit rusty. Through the years I have practiced every once in a while to remember the notes.
 * 2)What went poorly: Procrastinating on the research part of the project. I kept getting distrated by games and other homework, and thinking I'd have plenty of time to finish it. Which was a huge mistake that I often make and yet still can't get it into my head that procrastinating is really, really bad.
 * 3)Future strategies: Stop playing World of Warcraft, because it sucks the life out of you and is highly addictive. I have to do my work on a different computer so I don't see WoW on my desktop and get really tempted to log onto it.

title: User:DanielleFulgham


text:
User:Dannyhodonnell

User:Dannyhodonnell/Spanish

title: User:Dannyhodonnell


text:
Currents, and Circuits, and Variables, Oh My!

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Subject: Physical Science
 * }

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 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Sub-Subject: Electric Circuits, Scientific Method
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Length/Duration: 1 90-100 minute period
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#b4b4b4" style="border: 0px #cccccc solid;" |Technologies Used: Electric Circuit Simulator, Online Graphing, Moodle
 * }
 * }

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Relationship to Theoretical/ Empirical Findings
1. Throughout the design of the electric circuit lesson, students discussed as a class, gave examples of why they thought what they thought to each other, then finally were prompted to be specific in their learning of how they knew what they knew; a moodle forum prompt. Just as stated in an article by King (1994), if students practice lesson-based questions as well as ones intended to access prior knowledge/experience and promote connections between the lesson and that knowledge, their growth is that much more acknowledged. When students are reminded and made aware of their prior knowledge after having a collected body of evidence, they will be more likely to retain the new information learned. Why? Because one who has been in an experience is more likely to remember an integration of a familiar concept once experienced by self and a newly generated experience constructed within the classroom.

King, A. (1994). Children How to Question and How to ExplainGuiding Knowledge Construction in the Classroom: Effects of Teaching Children How to Question and How to Explain. American Educational Research Journal, 31(2), 338-368.

2. Because students will go out of their way (ignoring, shutting down, distraction) to avoid a fact that contradicts their current "logical" ideas about how something works, the classroom needs to establish a culture where a student may be called out at any moment to counter attack an idea that is against their norm. Although this relies more on the extrinsic motivation side, it seems to hold true that knowing you'll eventually be in an instance of defending a position against someone else's arguments will help increase instrinsic motivation due to increased competencies of practicing defending against other positions.

McGuire, W. (1961). The Relative Efficacy of Various Types of Prior-Belief Defense in Producimmunity against Persuasion. Journal of Abnormal and Social Psychology, 62(2), 327-337.

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Purpose and Background of Lesson
Purpose: Students explore multiple variables of electric circuits to collect data/ observations of how electric current (mA) and connections within a circuit are affected by those variables. Students also evaluate their thinking and how it has changed from beginning to end of the electric circuit activities.

Background: Students have built basic series circuits, parallel circuits, tested conductors and nonconductors within those circuits, and have explained what evidence there is to support that an interaction has occurred within these various elements of electricity. All of this was done through an inquiry based lab by following curricular procedures. As with most science concepts, students were asked to make a hypothesis to the following scenario before they began the above experimental investigations: “If you had 2 light bulbs within the same circuit, and both bulbs were shining light, what would happen to one of the light bulbs if you removed one of the working light-bulbs?”

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Evaluate/ Elaborate/ Extensions and Links
Evaluate: Teacher comments on moodle forum posts. Checks for evidence and process of thinking.

Elaborate: Student write a conclusion and reflection to their electric circuit activities for Homework. May use forum post for scaffold.

Extensions and Links

Other Insightful Educational Lesson Plans

Educational_Technology_Lesson_Plans

Educational Technologies

Educational_technologies

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Engage/ Explore/ Explain
Engage: See background.

Explore: Part 1: Students collect further data on how the number of cells within a circuit affects the amount of electric current that flows though that circuit. Students use an online Simulator: to run the experiment; students collect data by using an online ammeter attached to their circuits. The current (Ma) is measured for circuits using 1,2,3, then 4 cells incrementally.

Part 2: Student create a graph using an online program: Create a Graph. They use their values obtained from the 4 different cells and current. Once the graph is complete they upload it to moodle.

Explain: Students evaluate their process and changes in their thinking on an online forum. The forum allows students to post their thoughts so that the teacher and all others within the class can observe their ideas and also may respond to their ideas. The prompt/ purpose of the forum post is: “ Consider what you hypothesized as to what would happen to lights in a circuit if you took one out. Evaluate in detail ( 3 sentences minimum) how your thinking has changed now that you have evidence from our electric circuit lesson.” Students are required to look in their journals to find their observations/ evidence from today’s and the previous electric circuit interactions activities to make their evaluation credible and reflective as to what they thought at the beginning. Also, students are not required BUT were encouraged to reply to other student’s moodle forum posts on whether they agreed or disagreed with the other students thinking; if they disagreed, they were asked to give evidence from their personal information to counterargue the other student’s ideas.

 * }


text:
User:Dannyhodonnell/Introduction to Law 101

dannyhodonnell/introduction to law 101


text:
User:Rhodoraj

=Analytical Writing=

Personal Essay
Advertising I see it while I dress, while I eat, while I bathe and brush my teeth. It’s in my face while I go to school and even when I say hello to my friends. I’ve come to think that advertising is in everything. Whether it is good or bad advertising is another story.

I get up in the morning and the first thing I think of is how my breath doesn't seem so fresh. The only solution is to brush them. I stumble to the bathroom and apply toothpaste on my tooth brush as I read the huge colgate logo across the front of the tube. I wondered why I like colgate and just remebered not using any other toothpaste, plus it tastes good. I continue brushing and begin humming a fairly familiar tune unintentionally. Somewhat of an automatic reaction to brushing teeth.

I then hop in the shower continuing the melody as a fading hum. I start to scrub away and another tune involuntarily becomes audible but this time it comes out in full song. Then without me realizing "You're not fully clean unless you're zest fully clean!" echos off the tile of the shower wall. Visions of a waterfall in a rainforest and soap suds fill my head. This brings back memories of saved by the bell. Which I watched everyday after school in junior high. I believe that was the popular years of zest.

I dress up in my outfit for the day and I then look at what I'm wearing and wonder how it came to me to dress this particular way. I stand infront of the mirror and see that I am wearing three different patterns a high waisted skirt and a pair of boots. I get on my bike and continue down to school. On my way to school I count 15 girls all wearing boots. I also count 4 girls wearing high waisted skirts. Then I realize how these fashions all leaked into our lives in a way obvious or not. The fashions are being advertised in magazines ad as the "it" look for winter. As I enter school the numbers rise. Trends can be the most suttle yet very effective way of advertising and many people are not even aware that it is effecting them.

The new phone that a cool guy is on or the leather jacket that you see your bestfriend in. Advertising is all planned out but not all to the absolute detail. Advertising status and acceptance through trends and product is rather twisted but effective. Many people will agree that you dress for success, which means you would have to dress in clean and up to date clothing which means you would have to trade out old clothing after wearing them down to buy new clothing. Many people choose to purchase new clothing every season. Either to reinvent themselves, keeping status or keeping ahead of the trends. Peer pressure seems to be the most efficient type of advertising. I know that there will always be advertising and there will always be agressive or tactil ways but the most effective is peer and that's usually used through trends.

Children like most people want to fit in and no matter what spot they find in life it will effect them throughout the rest. Maintaining status as an outsider means more rebelious clothing or not as trendy clothes because no matter what there's an equation to everyone's madness or liking and there are many ways we express our status or demeanor. By advertising a type of person or a way of life there must be a version that pre exists. This is where trends begin. They begin in mass media whether it be a movie, a band, a tv show, or a moviestar. We all get effected by these visuals. Advertisements always find their way to the trends. Facebook, myspace, movie theaters, tv and music are only a small selection of advertisement in our daily life.

A majority of advertising is aimed toward children. This seems very tactical, considering most children want to be liked or a rebel or a bookworm or an athlete so why not advertise to children when they are the most avid soul seekers of our society. Insecurities and the need to express or fit in is swimming within their hormones.

I do believe that children are very sensitive to advertisement. Anything that seems fun, vibrant or happy definitely catches the eyes of a young child who has yet to experience anything that would dull that vision of theirs in any matter. So why not have a cartoon on a pack of cigarettes, the younger they start the longer they have to use the product, right?. Advertising is dangerous but it is everywhere and some of the advertising is for products we don’t see the potential of dangers until it is too late.

Smoking should not be advertised to children but what is the difference between having a parent that a child looks up to that smokes and a cartoon on a package that makes smoking look fun. It makes me wonder how many children start smoking because of someone in their family. Having just the right influence can change someones perspective on advertising and definitions of fun or cool. Which seem to be a huge drive for children. Where people fail the media will succeed. Television can gain the trust of children by just being turned on. Many children are raised by the television and therefore are exposed to hours of advertisments each week and many versions of cool or fun. Advertising whether direct or indirect will happen and parents need to become more dependable and understanding and patient in order to keep up with the growing strategies of advertisment. I believe that cartoon characters on cigarettes aren't going to be as effective if a child is informed by their parents. There are many other countries that put cartoons on everything. In Japan they put cartoons on condoms. This doesn't mean five year olds are having sex in Japan.

But as I said advertising is everywhere. It’s on what we eat, wear, get ready with and while we transfer from one place to another.Either indirect advertising vs. direct advertising, tradition (Your mother buys colgate so for the rest of your life you buy colgate), or peer pressure (“I buy what my friends buy”) Not all advertising is bad and not all trends either. We are actually in the beginnings of a earth friendly trend and it is bleeding it's way into corporates with everyday it becomes more popular. Possibly we could avoid our children making bad decisions if we educate them and allow them to choose And support their minds as individual rather than deciding for them.

Self Assessment
I've had Steve(n) Arntson as a teacher before and I can honestly say that enlightenment is in store by anyone who takes a class from him. Issues in everyday life get brought up and analyzed in ways they normally are not seen to the common college student. I've learned how to expand my resources not as a general view but also a personal view and how to break down big issues into the everday issues. Problems seem so unimaginably fixable but with the right research we can spread a common solution that everyone can partake. I do have much to learn as everyone does but I feel like I can be more confident in my endeavors with the insight on angles of research and being able to hon down on key issues.

I believe what I did well in this class was producing a different perspective in class and group discussions and I also approached my part of my group project with a different light. I brought up unique insights how reducing is the real key to our plastic issues. I know I could have communicate better. I could have discussed more with my group as well. I'm not very good at communicating my personal life but I am working on that. I let a lot of these issues get to me personally and I couldve been a little less sensitive to the matter.

I feel prepared for future projects because I feel more confident and aware of the resources and researching techniques that are required to throughly research a subject. I do feel unprepared on quoting and proper grammar. That I know is a weakness of mine and it always will be.

All in all this class has been very successful for me. I believe classes like these build character even if it may be a little hard on ones self to be exposed to real life at it's raw core.

title: User:Rhodoraj


text:
User:WriterProducer

WriterProducer
Washington, DC

Experienced writer, producer and director. 800 commercials, six documentaries, dozens of corporate, association and NGO projects, plus music videos.

Enhancing my feature film screenplay writing skills via Wikiversity and its creative community.

title: User:WriterProducer


text:
Template:LessonPlan

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Header 1
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Quiz
Below is a sample quiz. Edit the question and responses so that they pertain to your lesson plan.

{Type a question here... - Wrong answer - Wrong answer + CORRECT answer - Wrong answer
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{Type a question here... + CORRECT answer - Wrong answer - Wrong answer - Wrong answer
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{Type a question here... - Wrong answer + CORRECT answer - Wrong answer - Wrong answer
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Header 3
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Header 4
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title: Template:LessonPlan


text:
User talk:Arithmetix

 Hello Arithmetix, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 12 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 12 November 2009 (UTC)

title: User talk:Arithmetix


text:
User talk:Kayost09

 Hello Kayost09, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 12 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 12 November 2009 (UTC)

title: User talk:Kayost09


text:
User talk:Tkasper

 Hello Tkasper, 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

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 12 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 12 November 2009 (UTC)

title: User talk:Tkasper


text:
User talk:DefineUsername

 Hello DefineUsername, 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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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 12 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 12 November 2009 (UTC)

title: User talk:DefineUsername


text:
User talk:CatMad

 Hello CatMad, 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

 * Take a guided tour and learn to edit.
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 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 12 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 12 November 2009 (UTC)

title: User talk:CatMad


text:
Post-traumatic growth

Measures

 * Post Traumatic Growth Inventory (PTGI, Tedeschi & Calhourn, 1996;, , consists of 21 self-report items that assess perceptions of growth with respect to five aspects:
 * Relating to others
 * New possibilities
 * Personal strength
 * Spriritual changes
 * Appreciation of life
 * Questions ask about the extent to which change has been perceived to occur as a result of a traumatic experience to which participants respond using a six-point Likert scale.

title: Post-traumatic growth


text: File:5.C.Displacement Sensor.pdf

title: File:5.C.Displacement Sensor.pdf


title: File:3.A.DTFT.pdf


text:
Template:Findsources

<!--Place interwiki and category links on the documentation subpage, please.

title: Template:Findsources


text:
Template:Findsources/doc

Usage
This template produces a series of links to various Google resources to help find additional reference material for articles. The first parameter is main search term, which will be treated as a literal string if multi-part (e.g.  will be searched as the single string "Albert Einstein" not "Albert" and "Einstein" separately).

Optional subsequent parameters (up to 4) are additional search terms. Then can be enclosed in double quotation marks, preceded by an unspaced  (hyphen) to exclude them from the search results, or both (e.g.   to exclude matches containing the phrase "Marilyn Monroe", but not either name by itself, from the search results). You do not need to add, as the template automatically does this for you. Also, there is no need to add words in plural form unless irregular, as Google will match (for example) billiards as well given a search term of billiard.

If no parameters are given, the name of the article is passed as the search string. Note that this may or may not be desirable with disambiguated article names (if the Wikipedia disambiguator is sensible, however, it may actually help refine the search results).

If yes is added, Google News will only return free (as in beer) search results; please note that this can exclude many valuable sources simply because they have a small (or sometimes not so small) price tag; this parameter is best used when linking to source material in discussions, rather than when using the template near the top of an article's talk page as a source-finding resource.

 * Simple example:

 * produces:

 * Complex example:

 * produces:

Redirects

 * Find
 * FindSources
 * Find sources
 * Find Sources

zh:Template:Findsources

title: Template:Findsources/doc


text: Template:Para

title: Template:Para


text:
User:DonDon

Name: Don Gender: Male Age: 57 Location: Northwest Oregon, United States

title: User:DonDon


text:
User:DrJJ1967

I live in North Carolina with my beautiful wife, Denise, and our dogs Dutchess and Dr. Maximus (Max, for short). I have earned doctorates in Philosophy and Law and am attempting to pick up programming.

My interests include computer sciences, fitness, history, business, investing, real estate, philosophy, and social sciences.

I am a counselor, businessman, and teacher. My wife is ex-Navy and works in radiology.

I am particularly interested in COBOL, Java, Ajax, PHP, and all things pertaining to databases.

title: User:DrJJ1967