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).
6,636
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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?
...
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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
$$\nabla\cdot\Big [\boldsymbol{u}(\nabla\cdot\nabla p) - \nabla (\...
- 93
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25
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variational form of the functional
when reading some article, i have some problem to get the force $\mathbf{F}$ related to the energy functional from Hamilton's principle.
Given the functional:
$$
E=\frac{1}{2}k_{s}\int\left(\left|\...
- 21
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53
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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
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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
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0
answers
27
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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
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40
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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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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 ...
3
votes
2
answers
39
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Paramterizing the surface on the intersection of $x+z=a$ and interior of $x^2+y^2+z^2=a^2$
So I am trying to verify Stokes' theorem for $\vec{F}=y\hat{i}+z\hat{j}+x\hat{k}$ where the curve $C$ is on the intersection of $x+z=a$ and $x^2+y^2+z^2=a^2$. Solving these equations yields the curve $...
- 185
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2
answers
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Area Vector of a flat region
I don't understand the answers provided in the following problem (by the professor)
Let $\vec{v}=\vec{i}+8\vec{j}-7\vec{k}$ and S be the Rectangular region with the orientation shown below.
a) Find a ...
- 205
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0
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46
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The derivative of a function of several variables
Define a function $f:\mathbb{R}^2\to\mathbb{R}$ by: $$f(x,y)=\begin{cases}
\dfrac{x\sin y - y\sin x}{x^2+y^2}\ &\text{if}\ (x,y)\neq(0,0)\\0\ &\text{if}\ (x,y)=(0,0)
\end{cases}.$$
I want to ...
- 159
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vote
1
answer
45
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Derivation of the formula for parametrized surface area element
The surface integral is given by
$$\int_{S} f \,dS
= \iint_{D} f(\mathbf{\sigma}(u, v)) \left\|{\partial \mathbf{\sigma} \over \partial u}\times {\partial \mathbf{\sigma} \over \partial v}\right\| \,...
- 535
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1
answer
74
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Integrability of a vector field and its topology
Today, in the lecture, we covered an example of a vector field which suffices the necessary condition for integrability, yet is not integrable. The following field also known as the angular form is an ...
- 617
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1
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Expressions for directional derivative
I am reading a book called "Vector Analysis" by P.R. Ghosh and J.G.Chakravorty.
In it it is stated that-
'Consider a scalar point function $f(r)$ or $f(x,y,z)$ in the neighbourhood of the ...
- 33
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0
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Is $\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times \frac{\partial H}{\partial t}$?
While trying to prove a particular equation using Maxwell's equations in electromagnetic theory, there is a step in my textbook that says
$$\frac{\partial}{\partial{t}}(\nabla\times H) = \nabla \times ...
- 535