Show that for a gradient system $\bf\dot x= f(x)$, $\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}=0$ for $1 \leq i, j \leq d$
56936
$\frac{\partial f_i}{x_j}=\frac{\partial f_j}{x_i}\implies(f_1,\ldots,f_n)$ is a gradient
90287
How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$
45071
Having the derivative be an operator
133749
Trouble with gradient intuition
77331
$f$ function with $\nabla f=\bar{0}$
111470
Proof of differentiability of $f(x)$
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
19828
Show that $\frac{x^3}{x^2+y^2}$ is not differentiable at $(0,0)$, even though all directional derivatives exist
44761
Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$
75548
second derivative of the inverse function
129394
understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$
136714
Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$.
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
9003
prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation
103308
Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$
489
How to take the gradient of the quadratic form?
16644
Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?