I'm unsure how to do this problem:
(a + 2b)\nabla(\nabla \cdot \vec u) - b \nabla \times \nabla \times \vec u - (3a + 2b)c\nabla T(r)= \vec 0
\hat u = U_r \hat r + u_\theta \hat \theta +u_z \hat z
a,b,c constants
how would I solve this for u?
Thanks.
I've been confused because I've seen two forms of the polar laplacian:
1)\frac {\partial}{\partial r}( {1 \over r}\frac{\partial T r}{\partial r})
and
2) {1 \over r} \frac{\partial}{\partial r}(r \frac{\partial T}{\partial r})
how are these equivalent?
\nabla ^2 T = \frac {\partial}{\partial r}( {1 \over r}\frac{\partial rT}{\partial r})=0
is the polar form used,
which implies {1 \over r}\frac{\partial rT}{\partial r}= A and integrate to get my (incorrect) result above...
What am I missing!?
My result is :
T= \frac{1}{2} Ar + \frac{B}{ r}
A, B constants. Also in this example, I'm not sure how to apply the boundary conditions...
But the result I saw(without proof) involved logarithms...?
If I have a subset, how do I define an equivalence relation.
I understand it has to satisfy three properties:transitive, symmetric and reflexive, but I'm not sure how to give an explicit definition of the equivalence relation, for example on I where
I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}
Consider the heat equation in a radially symmetric sphere of radius unity:
u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty)
with boundary conditions \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0
Now, using separation of variables u=R(r)T(t) leads to the...
I have an integral \int \int_S x^2 + yz \ dS
and wish to transform to spherical polar coordinates. How does dS become
dS = r^2 \sin \theta d\theta d\phi ??
Where surface S is x^2 + y^2 + z^2 = 1
Thanks,
Q1-how would I evaluate:
G(y) = 1-F(2-y)\
\mbox{to give the cdf? Do I just substitute 2-y into }{1 \over 4} x^2 \mbox{ to give }
G(Y) = 1 - {1 \over 4} (2-y)^2 = 4y - {1 \over 4}y^2\ for \ 0 \leq y \leq 2
\mbox{for the cdf of Y? and so for the pdf: } 4-{1 \over 2}y
Q2-Why are...
To show it's odd:
look at values in the intervals?
-\sin(-\pi) - \sin({\pi }) = 0
\sin({-\pi \over 2}) + \sin({\pi \over 2}) = 0
-\sin({3 \pi \over 4}) - \sin({-3 \pi \over 4}) = 0
do I need to show anything else?
Hello
I'm not too sure if this is the correct location for my post, but it's the best fit I can see!
The cdf of the continuous random variable X is
F(x)=\left\{\begin{array}{cc}0&\mbox{ if }x< 0\\
{1\over 4} x^2 & \mbox{ if } 0 \leq x \leq 2\\
1 &\mbox{ if } x >2\end{array}\right...