# Tag Info

### Interpreting a notation in calculus of variations (differentiating with respect to a derivative)

Consider a functional $J[y]$ defined by: $$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$ I think it will be helpful to remind you right away that your notation implies that you have already agreed to ...
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### Proving function is differentiable at $(0,0)$ using total derivative definition

You can use that $\|h\|_2 = \sqrt{h_1^2+h_2^2}$ for $h\in\mathbb{R}^2$. If we now have a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ with $f(0)=0$ and we want to prove that $f'(0)=0$, this ...
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### Nonlinear homogeneous map $f : \Bbb{C} \times \Bbb{C} \to \Bbb{C}$

Consider an arbitrary continuous function $g$ on $\mathbb C$ st $g(0)=0, g(z)=z, |z| \ge 1, |g(z)| \le |z|$ but $g$ is not the identity. (eg take $g(re^{it})=r^2e^{it}, r \le 1, g(z)=z, |z| \ge 1$) ...
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### When does a smooth vector field induce an ODE?

Too long for a comment. A vector field $v$ always induces an autonomous first order ODE in two dimensions $$\tag{1} \pmatrix{\dot x\\\dot y}=v(x,y)\,.$$ I am not sure why that seems not what you want ...
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### Please explain Figure 24.4 in "Analysis on Manifolds" by James R. Munkres.

Yes, the blue surface is $A \cap M$ and the solid enclosed by the red closed curve $A \cap N$. The image of $F$ doesn't have to be rectangular, but you can always achieve this by restricting the ...
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