Standard for me would be to start with ## \frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}## and multiply by ##u## then integrate over the spatial domain ##[0,L]##.
I am not entirely sure how you defined ##[\mathbf{a},\mathbf{b},\mathbf{c}]##. Probably as a determinant? Either way, you must have ##[\mathbf{a},\mathbf{b}, \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})##. That might be helpful.
There is also an identity that relates...
You need to read up on the differences between scalars and vectors. You also need to read about cross products.
A scalar is just a (real) number.
A vector has both magnitude and direction. You should think of it as an arrow in the coordinate system. It has a certain length (its magnitude) and...
Do you know what a free index and a dummy index is?
How is ##\epsilon_{ijk}## defined?
What does ##A_{k,j}## look like?
If you are having trouble answering these questions, I suggest that you read up on it in your textbook or lecture notes.
As I said, write out each component of both expressions explicitly. You have more or less done so for ##\nabla \times \mathbf{A}##. It is probably more convenient to use ##\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3## rather than ##\mathbf{i}, \mathbf{j},\mathbf{k}##.
Can you write out each...
First of all, note that your ##i,j,k## mean different things in different equations. This could lead to some confusion.
Either way, I guess they want you to write out each component of ##\nabla \times \mathbf{A}## and ##\epsilon_{ijk}A_{k,j}## explicitly and verify that they are equal. Can...
Thank you, Pasmith.
It seems highly plausible that I was expected to identify the expression as the solution to that linear recurrence relation, which I assume is what you did by inspection.
Indeed, by considering ## b_n = a_n (3) ## I use the recurrence relation to find that ##b_n## is...
Homework Statement
From an old exam: Show that
\begin{equation*}
\sum_{0 \leq 2k \leq n} \binom{n}{2k}2^k = 0 (3) \text{ iff } n = 2 (4).
\end{equation*}
By ##a = b (k)## I mean that ##a## is congruent to ##b## modulo ##k##.
Homework Equations
Binomial theorem: ## (a + b)^m =...
The partial derivatives are not continuous. To see this, compute either of them at ##p \neq (0,0)## and take the limit as ##p \rightarrow 0##.
If I am not mistaken, what you showed is that the functions ## f_x(x,0), f_y(0,y)## of one variable (the other being fixed to zero) are continuous.
I don't see how you could have inferred any of those, or why you would need to.
You are asked to show that ##G## and ##T## are isomorphic. There is an obvious candidate for an isomorphism, so you should just verify that it actually is one.
Yes. I just wanted to rule out the possibility that the minimum occurred at an endpoint.
At the risk of being too obvious, say the planar curve ##\gamma(t) = (1+t, 0), \, t \in [0,1].##
Well, I am not that familiar with the theory, but you say yourself that you need to write your equation on the form
\begin{equation*}
\frac{dy}{dx} = F(\frac{y}{x}).
\end{equation*}
Does this seem doable?
That said, my first instinct would instead be to try the substitution ##u = x + y##.