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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2answers
32 views

Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products

On Wikipedia page on vector multiplication, they gave several ways to "multiply" two vectors, the most well known being the dot product and the cross product I am curious whether we can multiply two ...
1
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1answer
22 views

Finding Eigenvectors when we have lots of zeroes

\begin{array}{cc} 0 & 0 \\ 0 & 8 \\ \end{array} I have $\lambda_1=8$ and $\lambda_2=0$ but cannot find $V_1$ or $V_2$ I try \begin{array}{cc} \lambda & 0 \\ 0 & 8-\lambda \\ ...
2
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1answer
24 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
1
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1answer
19 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
0
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1answer
20 views

Surface Integral of the Partial Derivative of a Harmonic Function

Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$. Show that ...
1
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1answer
18 views

Existence of a Non-Linear Function Satisfying Certain Conditions

Let $f: \mathbb{R} \rightarrow \mathbb{R^m}$ and $N: \mathbb{R} \rightarrow \mathbb{R^m}$ be two mappings satisfying: $$\lim_{h\to0} \frac{f(a+h)-f(a)-N(h)}{h}=\vec 0.$$ If $f'(a)$ exists and is ...
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0answers
34 views

Using Gauss divergence theorem on cylinder

Use Gauss’s divergence theorem to compute $$\iint \limits_S F ·n \, \, dS $$ where $n$ is the outward normal for the following: (a) $S$ is the exterior surface of the cylinder $x^2 +y^2 ≤ 1$, and $0 ...
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1answer
17 views

Infinity as a boundary condition - Laplace's equation

I am familiar with the uniqueness theorem that claims that given conditions on the boundary of a closed surface, if there exists a function $\varphi$ such that $\nabla^2\varphi = 0$ which also satisfy ...
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0answers
12 views

What would the phrase “attain an upper bound of the line integral” mean for vector fields?

I am working on an exercise that is asking to find the upper bound of a line integral over the unit disk where the vector field has magnitude one. I am then asked to find a vector field that attains ...
2
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1answer
37 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
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2answers
28 views

linear transformation matrix under the line integral

Is there a general methodology/approach for evaluating an integral of this form? $$ \int_C {\bf Ax} \cdot \mathrm{d}{\bf x} $$ Assume ${\bf x} \in \mathbb{R}^{n \times 1}$ and ${\bf A} \in ...
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0answers
35 views

Find a vector field G such that F = curl(G), Given F = …

Given $F = (xe^y, -x \cos z, -ze^y)$, I find that $\operatorname{div}(F) = 0$ thus there should exist a vector field $G$ such that $\operatorname{curl}(G) = F$. I run into issues finding the solution. ...
1
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0answers
16 views

Stokes' theorem and line integrals

Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, ...
2
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1answer
61 views

Is the Laplacian a vector or a scalar?

Need to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is ...
-1
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0answers
26 views

Unanswered question that id like to know

I stumbled upon this question and have no idea how to solve it. Does anybody know how to solve it? $\oint_C \vec{F}\cdot d\vec{R}$ of $\vec{F}(x,y)= \begin{pmatrix} y^4+\cos x\\ x^2y \end{pmatrix}$
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0answers
16 views

Intersection of curves and constructing a plane

Can someone please help me with how to approach/solve this question? Show that the following pair of curves intersect, and construct a plane that is tangent to both curves at the point of ...
0
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1answer
27 views

$\oint_C \vec{F}\cdot d\vec{R}$ of $\vec{F}(x,y)= \begin{pmatrix} y^4+\cos x\\ x^2y \end{pmatrix}$ [closed]

If $C$ is the curve in the first quadrant over the line $y=1/3\sqrt{3}x$, the arc $x^2+y^2=1$, the line $y=x$ and the arc $x^2+y^2=2$ and $\vec{F}$ is the vectorfield $\vec{F}(x,y)= \begin{pmatrix} ...
0
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0answers
20 views

Limits of this parametrisation

Use Stokes’ theorem to solve the following integral (each time the curve is oriented counterclockwise when viewed from above). $$\int \limits_C (x+2y)dx+(2z+2x)dy+(z+y)dz$$ where $C$ is the ...
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1answer
28 views

Area of surface [duplicate]

What is the area of the region that is bounded by the curve $$\vec{R}(t)=(\cos^3t, \sin^3t), 0\leq t<2\pi?$$ I have no idea how to start here or what i have to use.
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1answer
35 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
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0answers
27 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
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1answer
22 views

Divergence of $\phi$ from p

I am reading a paper which is based mostly on divergence. I tried to get a basic understanding of divergence but I cannot see how it is linked with this aspect. It says: $D(\phi,p) = \phi . ...
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votes
4answers
33 views

Difference between $\nabla T$ and $\nabla \cdot E$

Why is $\nabla T = (\frac{\delta T}{\delta x},\frac{\delta T}{\delta y},\frac{\delta T}{\delta z})$, but $\nabla \cdot E \neq (\frac{\delta E}{\delta x},\frac{\delta E}{\delta y},\frac{\delta ...
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votes
3answers
44 views

Why is $a\times b = 0$ when $b = 2*a$?

In vector calculus, why is $a\times b = 0$ when you know that $b=2*a$? So how how do you know that a crossproduct of a vector and two times that vector is always zero?
0
votes
1answer
19 views

Integral of 2-D Laplacian

I am so confused on these integrals. Here is the question. Problem $$G(x,y)=\ln(x^2+y^2)/2$$ Calculate the 2-D Laplacian $\Delta^2G$ For the interior $D$ of the circle $C$ of radius $a$ calculate ...
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0answers
12 views

Calculating the Flux of a Surface

I am having some trouble with this problem, I feel like I am just confusing myself and I could really use some direction. Problem "For positive $a$ and $h$ let $A$ designate the region of $R^3$ ...
1
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1answer
12 views

Under what condition does $A^T(B \times C) + (B\times C)^T A = 2A^T(B \times C)$, A,B,C vectors

In my classical mechanics text book there is a formula that states $(\dot r_c + \omega_i \times d_i)^T (\dot r_c + \omega_i \times d_i)$ give rise to $\dot r_c^T \dot r_c + 2\dot r_c^T(\omega_i ...
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0answers
20 views

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$ I changed $x^2+y^2=4$ to $4sin^2\theta + 4cos^2\theta = 4$ so $x=2cos\theta$ and $y=2sin\theta$ and then ...
2
votes
1answer
21 views

Closed surface integrals

Can somebody give me hints to solve the following question? I need to find the closed surface integral (using divergence theorem) of $$\oint \vec{r} (\vec{a} \cdot \vec{n}) da$$ where $\vec{n}$ is ...
2
votes
2answers
19 views

What will be value of $\vec{r} \cdot \nabla$

I was studying on Nabla Operator and saw that $\nabla \cdot \vec{r} \neq \vec{r} \cdot \nabla$ So, if I were to find $\vec{r} \cdot \nabla$ how would I calculate it? I know that $\vec{r} \cdot ...
2
votes
2answers
33 views

Divergence and Curl (involving constant vectors)

How find the divergence and Curl of the following: $(\vec{a} \cdot \vec{r}) \vec{b}$, where $\vec{a}$ and $\vec{b}$ are the constant vectors and $\vec{r}$ is the radius vector. I have tried solving ...
3
votes
1answer
22 views

Divergence and Curl of the vectors

How to find the divergence and the curl of the given vectors? a. $( \vec{u} \cdot \vec{r}) \vec{v}$ b. $( \vec{u} \cdot \vec{r}) \vec{r}$ c. $( \vec{u} \times \vec{r})$ d. $ \vec{r} \times(\vec{u} ...
2
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0answers
19 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
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0answers
9 views

Derivation of centrifugal acceleration with Coriolis theorem

Is there a way to derive the centrifugal acceleration of an object rotating with a constant speed $V$ on a circle with radios $r$ with Coriolis theorem ?? thanks
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0answers
34 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
1
vote
1answer
16 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
4
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1answer
43 views

If $\alpha$ is a unit speed curve of constant curvature lying in a sphere, then $\alpha$ is a circle.

I'm trying to solve the following problem but got stuck along the way. I would like some help on getting this through. Prove that if $\alpha$ is a unit speed curve of constant curvature lying in a ...
0
votes
2answers
27 views

If $\vec p\times((\vec x-\vec q)\times\vec p)+\vec q\times((\vec x-\vec r)\times\vec q)+\vec r\times((\vec x-\vec p)\times\vec r)=0$, find $\vec x$

Let $\vec p, \vec q$ and $\vec r$ are three mutually perpendicular vectors of the same magnitude. If a vector $\vec x$ satisfies the equation $\begin{aligned} \vec p \times ((\vec x - \vec q) \times ...
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votes
3answers
63 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
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3answers
26 views

Finding a point a certain distance away from 2 points

I need to find a point that is a certain distance away from two known points. Where $P_1, P_2, L_2$ and $L_1$ are all defined and that is all that is known. How do I find $P_3?$ Kind Regards.
2
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1answer
27 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
0
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0answers
21 views

the integral of normal derivative

I am studying Kreyszig's Advanced Engineering book section 10.8. I am been trying to solve this exercise, but I am not sure. Could you please help. I want to calculate $\iint_S (\partial f/\partial ...
1
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2answers
37 views

Question in vector algebra regarding minimum value of modulus.

If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three coplanar unit vectors such that $\vec{a} +\vec{b} +\vec{c} =0$. If three vectors $\vec{p}$ , $\vec{q}$ , $\vec{r}$ are parallel to $\vec{a}$ , ...
0
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0answers
19 views

Using Stokes's theorem

Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, ...
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0answers
18 views

How to do surface integral in spherical co-ordinates?

These are in spherical co-ordinates.How can I do surface integral in spherical co-ordinates? Do I have to change them in x,y,z co-ordinate and do the surface integral or is there any other way? ...
1
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1answer
43 views

What are some good books on vector analysis in higher dimenesion

What is some good books specifically on vector analysis in higher dimension? Standard vector calculus book usually only introduced double and triple integral method
0
votes
1answer
34 views

using the divergence theorem

Use Gauss Divergence Theorem to compute $$\int \int \limits_S F\cdot n \,dS$$ where $n$ is the outward normal for the following: $S$ is the surface $x^2 +y^2=z^2$ and $z \in [0,1]$, and ...
1
vote
1answer
36 views

Convert a general equation system to matrix form

I have an equation, where $x$ is a vector of variables and the rest (vector $a$, vector $b$, matrix $M$ and $c_1, c_2, c_3$) are parameters: $a_i - x_i + c_1\sum_{j=1}^{n} m_{ij} x_j - c_2 ...
34
votes
3answers
547 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
8
votes
3answers
80 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...