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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1answer
39 views

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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26 views

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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1answer
15 views

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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0answers
23 views

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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2answers
33 views

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
-1
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1answer
25 views

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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0answers
14 views

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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1answer
10 views

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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0answers
38 views

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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1answer
17 views

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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0answers
91 views

A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\...
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1answer
69 views

Stoke's Theorem Theory Question

If $C$ is the boundary of a surface $S$ and $\phi$ and $\psi$ are arbitrary smooth scalar fields, Show that $$\int_C \phi \nabla \psi \cdot dr = - \int_C \psi \nabla \phi \cdot d r = \iint_S (\nabla \...
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0answers
26 views

Exist this vectorial equality, and this is correct?

doing a problem about distance with vectors appears this identity: $$\vec{A}\times\vec{B}=\frac{\vec{A}(\vec{A}\cdot\vec{B})-\vec{A}^{2}\vec{B}}{\lvert\vec{A}\lvert}=(\vec{A}\cdot\vec{B})\hat{A}-\...
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1answer
43 views

Vortex flow - Surface Integral

Consider the vortex flow of a fluid of density $\rho$ where the fluid rotates with an angular velocity $\omega$ about the $z$-axis. Determine where a unit square $S$ on the $yz$-plane should be placed ...
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1answer
19 views

If the integral over an area is zero is the integral of the gradient also zero?

Say I know that $\int\int v_z dx dy = 0$ over some area with $dA = dx dy$. $v_z$ is a function of $x$ that "points" in $z$. Is this enough to say that $\int\int \frac{\partial v_z}{\partial x} dx dy = ...
3
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1answer
32 views

Geometric intuition of the equation of a plane

Let $\pi$ be a plane in an $d$-dimensional space with normal vector $\underline{w} = [w_1, \dots,w_d]^T$. If point $\underline{p} = [p_1, \dots,p_d]^T$ is in the plane and $\underline{x}= = [x_1, \...
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1answer
26 views

Tangent plane of a surface at points with given gradient

So I'm stuck on the next problem: I need to find the tangent plane of the surface $$u=\ln\left( x+\frac{1}{y} \right)$$ at all the points where the gradient is equal to $$\nabla u=\hat i-\frac{16}{9}\...
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1answer
22 views

Symmetry of Green's function on the general case

Let's consider the differential equation $$\nabla\cdot[p(\mathbf{r})\nabla u(\mathbf{r})]-s(\mathbf{r})u(\mathbf{r})=-f(\mathbf{r}).$$ I want to show that the Green's function is symmetric, so that $...
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0answers
38 views

Question about simply connected regions in $\mathbb{R}^2$

If given the vector field $\mathbf{F}(x,y)=\langle\frac{1}{x}+2xy,\frac{1}{y}+x^2-\cos{y}\rangle$, it is clear that the component functions are continuous everywhere expect at the origin. Now, when ...
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1answer
147 views

Relationship between Möbius transformations and flows/vector fields

I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For ...
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22 views

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $ where $\sigma$ is the surface in the first octant made up of part of the plane $2x+3y+4z=12$ and triangular in the $(x,z)$ ...
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1answer
33 views

Find a closed path $C$ such that $\oint_C {\bf F} \cdot d{\bf r} \neq 0$, where $F = (y^2,x,0)$

Consider the vector field $${\bf F}=(y^2,x,0).$$ Find a closed path $C$ such that $$\oint_C {\bf F} \cdot d{\bf r} \neq 0 .$$ My attempt: I decided to try with the unit circle however the ...
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0answers
16 views

How to take partial derivative of a vector matrix vector multiplication?

I am trying to understand the mechanics of the below equations. I am especially confused about in 2.65 , how did the r.h.s which is a sum came from the gradient vector ? It would be great if someone ...
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2answers
31 views

Differentiation of $x^TAx$

I have in my text that if I differentiate $x^TAx$ with respect to the vector $x$ I get $2xA$ - could I ask why? Here $x$ is a $3\times1$ vector, $A$ is a $3\times 3$ matrix - I am given the ...
3
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1answer
83 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
0
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1answer
30 views

Gradient Chain Rule: Applying Gradient in the case of a Series of Matrix operations (Neural Net Gradient Calculation)

I have the following situation: I need to calculate the gradient of the Error of a CNN a few layers deep by hand. Starting with the Error function, The $\operatorname{Error}[readoutX]= -\sum_i \...
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1answer
19 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
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1answer
26 views

Find the curl of the vector field G

Find the curl of the vector field: $\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$ where $r=(x^2+y^2)^\frac{1}{2}$ Since r is in the vector field, does it require calculation ...
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1answer
59 views

Solving constrained Euler-Lagrange equations with Lagrange Multipliers (Geodesics)

I'm trying to solve a calculus of variations geodesics problem using Lagrange Multipliers, showing that the geodesics of a sphere are the so-called great circles. I am using a constrained Lagrangian ...
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0answers
20 views

Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
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1answer
26 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. $$...
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1answer
36 views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) $\delta_{i\,j}...
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23 views

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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0answers
20 views

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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0answers
27 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define $\phi(\textbf{p})=\displaystyle\iiint_W\frac{\rho(\...
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43 views

Volume enclosed by Implicit Surface

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta \...
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1answer
37 views

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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1answer
59 views

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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75 views

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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1answer
35 views

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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0answers
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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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2answers
32 views

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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1answer
20 views

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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1answer
58 views

Outward flux of a vector field through a rectangular box.

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+...
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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?
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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 ...
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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 $x_0$, ...
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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 ...
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0answers
15 views

Approximating the line integral

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 ...
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1answer
39 views

Proving ${\displaystyle{\int\!\!\int_{D}\!\!u\Delta udA<0}}$ where $D$ is the closed, unit disc.

Hello again guys and gals! I'm stuck on a problem that I thought was going to be simple for me to prove, but I was wrong (yet again). I will try not to be so long-winded this time (see my previous ...