All Questions
Tagged with vector-analysis solution-verification
106
questions
2
votes
1
answer
70
views
Limits of triple integral over a tetrahedron.
The tetrahedron has vertices $O(0,0,0); A(0,0,2); B(0,2,0); C(1,0,0)$
I was thinking that the plane $ABC$ has equation $2x+y+z=2$ since:
$\vec{BA}= \langle 0,-2,2 \rangle$ and $\vec{CA}= \langle -1,0,...
0
votes
1
answer
103
views
What is the Divergence of a Spherically Symmetric Vector Fields?
A vector field is spherically symmetric about the origin if, on every sphere centered at the
origin, it has constant magnitude and points either away from or toward the origin. A vector
field that is ...
- 4,450
2
votes
0
answers
52
views
Verify (or critique) this informal proof of Green's theorem
In order to better understand Green's Theorem, I developed this informal proof, which I request verification and critique of (both the proof and its writing). Of course, any textbook has a proof: my ...
- 4,450
0
votes
0
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25
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Relationship between the line with gradient -1 and (-1/3) issue with tan(theta)=m argument - PLEASE HELP!
I was tutoring a student today and they asked a very interesting question while we were looking at the $tan(\theta)=m$ formula for the angle of inclination/the angle with the positive direction of the ...
0
votes
0
answers
38
views
Prove that in any conservative vector field $\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$
Let $F$ be a conservative vector field (smooth and defined on all $\mathbb R^2$), that is, $\int_P F \cdot dr$ depends only on the start and end of path $P$. Then at every point $\frac {\partial F_y}{...
- 4,450
0
votes
1
answer
43
views
Do vector calculus identities prove symmetries in Euler-Lagrange equations?
Let $\mathrm{S}$ be a differentiable manifold, and $T\mathrm{S}$ its tangent bundle. Let $\mathcal{L}:\mathbb{R}\times T\mathrm{S}\rightarrow \mathbb{R}$ be a Lagrangian functional $\mathcal{L}=\...
0
votes
0
answers
36
views
Prove there exists a $P$ such that $\nabla P = F$ iff $(\frac{\partial F}{\partial x})_y = (\frac{\partial F}{\partial y})_x$
Let $F: \mathbb R^2 \to \mathbb R^2, X: \mathbb R^2 \to \mathbb R, Y: \mathbb R^2 \to \mathbb R$ be smooth functions such that $$f(x,y) = \begin{bmatrix}X(x,y) \\ Y(x,y) \end{bmatrix}.$$ Show that ...
- 4,450
3
votes
1
answer
334
views
Prove Kepler's second law of planetary motion
An object moves in $\mathbb R^3$ it's position $r(t)$ satisfies $$r''(t) = s(t)r(t)$$ for some scalar function $s$ (a central force field, in which all acceleration is directly towards or opposite the ...
- 4,450
1
vote
1
answer
68
views
Determine the flux of $u=(xz,yz,z^3)$ out of unit sphere $x^2+y^2+z^2=1$ (Verification)
Determine the flux of $u=(xz,yz,z^3)$ out of the unit sphere $x^2+y^2+z^2=1$
$\textbf{Solution}$: I have no idea how this solution can be wrong but apparently it is, By Stoke's theorem, instead of ...
- 516
0
votes
0
answers
64
views
Vector field is conservative: proof verification
I've been trying to prove that this vector field:
$$
\vec{F}=\left(\frac{y}{\left(x-1\right)^{2}+y^{2}},\frac{1-x}{\left(x-1\right)^{2}+y^{2}}\right)
$$
Is conservative in:
$$
D=\left\{ \left(x,y\...
- 87
0
votes
1
answer
65
views
Computing the tangent plane of a surface $f(x,y,z)=0$
Let $\alpha\in\mathbb{R}$ be fixed and let $f:\mathbb{R}^3\to\mathbb{R}$ be the (smooth) function $f(x,y,z)=(x^2+\alpha y^2)e^z-2\alpha$. If $\alpha>0$ note that the point $\bf{p}=(\alpha,\sqrt{\...
user1022331
0
votes
1
answer
169
views
In two dimensions show that the divergence transforms as a scalar under rotations.
The following problem is from the Vector Analysis Chapter 1 of Griffiths' Electrodynamics
1.17 In two dimensions show that the divergence transforms as a scalar under rotations.
A question has been ...
- 5,021
0
votes
1
answer
42
views
What are the justifications for the steps involved finding the potential of a vector field by line integrating the latter?
There are a series of problems in Apostol's Calculus where we are given a specific vector field $\pmb{f}$ and are asked to determine if the vector field is a gradient of a scalar field, ie potential ...
- 5,021
0
votes
0
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69
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Question on why gradient of a suitable function is orthogonal to level sets
Let $f:\mathbb{R}^n\to\mathbb{R}$ be nice-enough function, say at least $C^1$. The argument I saw during my multivariable calculus course on why the gradient of $f$ is orthogonal to a given level set ...
- 1,221
1
vote
1
answer
246
views
Prove vectorially that the perpendicular bisectors of the sides of a triangle are concurrent
How to prove (vectorially) that the perpendicular bisectors of the sides of a triangle are concurrent?
My Attempt:
Let $\triangle OAB$ be our triangle, let us take the positive $x$-axis along the ...
- 26.9k