Questions tagged [vector-analysis]
Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).
6,630
questions
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What operation on matrices corresponds to the curl of a vector field?
Given the total derivative $Df$ of a (sufficiently) smooth function $f:\mathbb{R}^n \to \mathbb{R}^n$, the trace of the total derivative matrix corresponds to the divergence of $f$ (considered as a ...
- 1,458
3
votes
1
answer
64
views
How do I make my formula for $\nabla \times \mathbf{F}(x,y)$ correct?
Apparently, the curl of a vector field is a function that outputs the "rotationality" of the vector field at some point, as a function of that point's coordinates. I want to go from this ...
- 1,129
0
votes
0
answers
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views
Finding a smooth surface with boundary as a Jordan curve in a simply connected domain
Consider the following:
For any smooth Jordan curve $\gamma\subset D$, where $D$ is simply connected, we can find a smooth surface $\Sigma \subset D$ such that $\partial\Sigma = \gamma$.
I am curious ...
- 617
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votes
0
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Confusion about orientation of line integral
Consider a continuous curve $\gamma$ in $\mathbb{R}^3$ parameterized by $\mathbf{r}(t)$ as $t:a \rightarrow b$. Now, it is my understanding that line integrals
$$
\int_\gamma \phi \ ds
$$
for a scalar ...
- 29
0
votes
0
answers
24
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complex step derivative for gradient with complex values variables
Can the complex step derivative (https://mdolab.engin.umich.edu/wiki/guide-complex-step-derivative-approximation) be used for functions of complex variables ?
$f: \mathbb{C^n} \mapsto \mathbb{C^n} $ ...
- 113
1
vote
0
answers
39
views
What is the surface integral?
Let $u=((x+y)z,y+xz,z^2+xz)$ be a vector field and $\Sigma$ be the surface $\frac{x^2}{9}+\frac{y^2}{4}+\frac{z^2}{36}=1$. Calculate $\iint_{\Sigma}(u\cdot N)dS$.
So $\text{div }u=1+3z+x$ and by using ...
- 857
1
vote
1
answer
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views
How do you know when level curves are straight?
I have a smooth function $F(x,y)$ which sends an open disc $D\subset \mathbb{R}^2$ in the plane to $\mathbb{R}$. I would like to know how to compute whether the level curves of $F$ are all each (a ...
- 17.7k
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votes
2
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views
Curve is traveled clockwise or anti-clockwise
Given the curve
$$
\vec{\mathbf{r}}(t)
= \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}}
+ (4 - t^2) \, \vec{\mathbf{j}},
$$
how can I tell whether it's traveled clockwise or counterclockwise?
...
1
vote
0
answers
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views
What is the Laplacian of the Gradient? Is $\boldsymbol{u}(\nabla\cdot\nabla p) = \nabla (\boldsymbol{u}\cdot\nabla p)$?
I am supposed to find out whether for a
scalar function $p$ and a
divergence-free vector function $\boldsymbol{u}$
we have that
$$\nabla\cdot\Big [\boldsymbol{u}(\nabla\cdot\nabla p) - \nabla (\...
- 93
0
votes
0
answers
24
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variational form of the functional
when reading some article, i have some problem to get the force $\mathbf{F}$ related to the energy functional from Hamilton's principle.
Given the functional:
$$
E=\frac{1}{2}k_{s}\int\left(\left|\...
- 21
0
votes
0
answers
53
views
Struggling to understand vector calculus
I was working through an energy calculation for the Fokker-Planck equation:
$$
\partial_t \rho + \nabla\cdot(\mathbf{v}\rho) = 0
$$where
$$
\mathbf{v} = -(\frac{1}{\rho}\nabla\rho + \nabla V).
$$
At ...
1
vote
1
answer
46
views
Domain $U$ for "if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative"
$U$ is the union of two disjoint open simply connected sets. $\mathbf{F}:U\to\mathbb{R}^3$ is $C^1$. Then is it true that if $\nabla\times \mathbf{F}=\mathbf{0}$, then $\mathbf{F}$ is conservative?
...
user1099762
2
votes
0
answers
27
views
Stokes theorem to calculate line integral
Let $\gamma$ be the intersection between $z=x^2+y^2$ and the plane $z=1+2x$. Calculate the work done by the field $F=(0,x,-y)$ when the curve $\gamma$ traverses on lap in positive direction seen from ...
- 857
0
votes
0
answers
40
views
Calculus Identities
I am trying to write an expression to $\partial_t \|\nabla u\|_{L^p(\Omega)}^p.$ Here $\Omega$ is a smooth domain, the function $u$ has no regularity problems (you can take it smooth) and the normal ...
- 343
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votes
0
answers
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Problem with a proof on a vector calculus book
I have seen a proof that concludes this:
$\iiint_{V} \nabla \times \mathbf{B} \, dV = \iint_{S} \mathbf{n} \times \mathbf{B}\,dS$
My question is: if is it possible to take the volume integral of a ...
