# Tagged Questions

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).

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### Circle as oriented $1$-simplex and it's boundary

Let $\gamma(t)=(r\cos t, r\sin t)$ where $r>0$ is fixed and $t\in [0,2\pi]$. Rudin write that it's an "oriented 1-simplex". Also he states that $$\partial \gamma=0.$$ Let $T(u)=(r\cos u,r\sin u)$ ...
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### How do I indicate this identity?

Here,I want to show the following. \begin{align} I &=\int(\nabla\times\textbf{h})\cdot (\nabla\times\delta \textbf{h})d\textbf{r} \\ &= \int\nabla \times \nabla \times \textbf{h} \cdot\delta \...
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### Surface integral of Gaussian curvature on Torus

How to calculate the integral $$\iint_{\mathbb{T}} K \, dA,$$ where $\mathbb{T}$ is the torus with $R$ being the distance from the center of the torus to the center of the tube and $r$ is the ...
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### Find $F'(r)$ of $F(r)= \iiint_{x^2+y^2+z^2 \leq r^2} f(x,y,z) \, dx \, dy \, dz$

I have got the following problem. Let $f: \mathbb{R}^3 \to \mathbb{R}$ and $F: \mathbb{R}^+ \to \mathbb{R}$. $$F(r)= \iiint_{x^2+y^2+z^2 \leq r^2} f(x,y,z) \, dx \, dy \, dz$$ How to find $F'(r)$? ...
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### Evaluate $\int \int _S (x^2+y^2) dS$ where S is the surface $z=4-x\; 0 \leq x \leq 2\;\; 0\leq y \leq 2$

Evaluate $\int \int _S (x^2+y^2) dS$ where S is the surface $z=4-x\; 0 \leq x \leq 2\;\; 0\leq y \leq 2$ can we find the integral with using x and y but what about the z=4-x
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### Finding an expression for velocity [closed]

Consider an annulus formed by two circular cylinders, with one cylinder inside the other. The inner cylinder has radius $a$ and the outer cylinder has radius $b$. The cylinders have a common axis, and ...
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### Potential of vector field in spherical coordinates

I can't find any information about finding the potential for vector field, using the spherical coordinates. The vector is in form $\textbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$. I would be very glad for ...
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### Show that $\int r\cdot n ds$ equals three time the volume of $\omega$.

Let $\Omega$ be an open region in $\mathbb{R}^3$ with surface $∂\Omega$ on every point $P$ of which the unit outward pointing normal $n = n(P)$ is well defined and smoothly varying. Let $r = (x, y, z)$...
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### Stokes' Theorem from PMA Rudin. Confusing moment with simplex

I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may ...
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### Vector calculus and parameterising line integrals

Verify Stokes’s theorem for $F=z^2\mathbf{i}+5x\mathbf{j}$ and $S: 0\le x\le1,\; 0\le y\le1,\; z=1,\,$ where $C$ is the closed curve enclosing the surface $S$. I know how to compute Stokes ...
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### Necessity of $C^{1}$ hypothesis in fundamental theorem for line integrals

The statement for the fundamental theorem for line integrals I have in my (unpublished) textbook is: Let U ⊆ Rn be an open set, let φ : [a,b] → U be a piecewise smooth curve, and let $Ω = C_{φ}$. Let ...
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### Surface integral of prism

I have a prism bounded by x=0, y=0, y=1-x, z=0 and z=2, and the field $v=(3x^2,xy^2,0)$ and i want to find the flow rate out of this prism. I've already figured out that only the side on y=1-x is not ...
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### Stoke's Theorem to evaluate line integral of cylinder-plane intersection

I want to use Stokes' Theorem to evaluate the line integral $F\cdot dr$ $F = (-y^2, x, z^2)$ and $C$ is the curve of the intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1$. $C$ should be ...
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### Show that ∇· (∇ x F) = 0 for any vector field [duplicate]

To solve this question, how do I define any vector field $F$, in order to solve it? I called $F = (ax,by,cz)$, in which case already $\nabla\times F = 0$. How would i go about proving this? Many ...
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### Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
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### Using Stokes' Theorem to find the line integral

I am having a bit of trouble understanding line integrals. I've muddled my way through a lot of them, but I just can't understand their relation to Stokes' theorem. Here is a question that I've ...
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### Calculation of Variational Derivative

Following is from Olver's book on Lie groups and Differential Equations: Define the variational problem: \begin{eqnarray*} \mathcal{L}= \int_\Omega L(x,u^{(n)}) \end{eqnarray*} where $u^{(n)}$ ...
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### What does it mean to use levi civita symbol with Poisson brackets in this way

I'm doing some studies in mathematical methods for physics and I came across something that I don't really understand. I have only been using the $\epsilon_{ijk}$ when I cross some vectors or ...
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### Function Composition, Derivatives, Gradient, Hessian

Here's the problem: Let $f : R^n \to R$ be a twice continuously differentiable function. Let $\phi(t) = f(u + td)$ be a composition function from $R$ to $R$, with given vectors $u, d \in R^n$. ...
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Let $R$ be the rectangular box consisting of all points $(x,y,z)$ with $-1\leq x\leq 1$, $-1\leq y\leq 1$, and $-1\leq z\leq 1$. Define the vector field $$V= x(x^2+y^2+z^2)^{-3/2}\mathbf{i}+y(x^2+y^2+... 0answers 25 views ### Is the Divergence of Curl equal to Zero for All Coordinate Systems? Is the divergence of curl equal to zero for all coordinate systems? Even a curvilinear coordinate system such as double spheroidal coordinates? 1answer 90 views ### What is the most general/abstract way to think about Tensors In their most general and abstract definitions as Mathematical Objects : A Scalar is an element of a field used to define Vector Spaces A Vector is an element of a Vector Space. Since a Scalar ... 0answers 27 views ### Show that the Hessian of f is negative definite. Problem says: Show that if f:A\subset \mathbb R^2 \rightarrow \mathbb R has a critical point x_0 \in A and we let \Delta =determinant of Hessian of f be evaluated at x_0, ... 0answers 23 views ### Finding Directional Derivatives with gradient Find the derivative of the function at P_0 in the direction of u.$$f(x,\, y,\,z) = \tan^{-1}\left ( \frac{5x}{9y+2z} \right ),\,\,\, P_0(7,\,0,\,0),\,\,\, u = 12i - 3j+4k I understand how to ...
I am solving a series of problems that begins with, suppose curl $\vec{F}=\langle 5,4y,-2z\rangle$ and $C$ a circle of radius .005 centered at (2,4,5) in the plane $x+y+z=11$. The first part of the ...