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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2
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1answer
46 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 ...
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2answers
41 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 $x^2+y^2+z^2=1$, ...
2
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1answer
72 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
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1answer
37 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
45 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 ...
36
votes
3answers
625 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
86 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 ...
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votes
2answers
43 views

Solution of an integral containing vectors

I'm currently trying to solve the integral: $$ I(\vec{a},\vec{b})=4\pi\int\limits_0^1\frac{\mathrm{d}u}{1-(\vec{a}u+\vec{b}(1-u))^2}, $$ but I can't seem to find a good starting point. I know that if ...
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0answers
40 views

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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0answers
46 views

Simple Vector Calculus Integral

A little stuck on what I assume is probably a fairly basic vector calculus integration problem, but I haven't been able to find any resources online that deal with multivariable integration in a way ...
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0answers
31 views

Derive the equation of first variation for a flow of a vector field.

This is a problem from Susan Colley's Vector Calculus. I have trouble understanding the solution to it. Problem: Derive the equation of first variation for a flow of a vector field. That is, if ...
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2answers
46 views

Wording on this curl question

Consider the scalar field defined below: $$f:\mathbb{R}^3 \rightarrow \mathbb{R}^3, \hspace{2mm} F(x,y,z)=(x^2y^3,xy,xz^4)$$ Find the curl of $f$ at each point where it exists. I am a bit confused on ...
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2answers
14 views

cuboid with $z=0$ and $z=y$

Compute $\int \int _S F \cdot n \hspace{2mm} dS$ where $$F(x,y,z)=(x-z\cos y, y-x^2+x\sin z+z^3, x+y+z)$$ and $r$ is the surface that bounds the solid between the planes ...
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0answers
21 views

divergence theorem cube question

Compute $$\int \int _S F \cdot n \hspace{2mm} dS$$ where $S$ is the surface of the cube bounded by the six planes $$x=0,\hspace{2mm}x=2,\hspace{2mm}y=0,\hspace{2mm}y=4,\hspace{2mm}z=0, \hspace{2mm} ...
0
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2answers
48 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
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vote
3answers
62 views

vector field question

Consider the vector field $$F(x,y,z)=(zy+\sin x, zx-2y, yx-z)$$ (a) Is there a scalar field $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ whose gradient is $F$? (b) Compute $\int _C F\cdot dr \neq 0$ where ...
0
votes
1answer
38 views

the position vector $x(t_0)$ is orthogonal to the velocity vector $x'(t_0)$ if $x(t_0)$ is the point on the image of $x$ closest to the origin .

Let $x(t)$ be a path of class $C^1$ that does not pass through the origin in $R^3$. If $x(t_0)$ is the point on the image of $x$ closest to the origin and $x'(t_0)\neq 0$, show that the position ...
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0answers
20 views

cylindrical and spherical coordinates

This is a very hard question to explain. In vector analysis, when dealing with surfaces, stokes theorem, gauss div theorem, etc. The cylindrical coordinates are: $x=r\cos\theta$ $ $ y=r\sin\theta$ ...
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1answer
35 views

Law of Cosines, Trigonometric Angle Addition Theorems, and Dot Product Relations

Just as the derivative, slope, and gradient are essentially the same thing I've realized that the Law of Cosines, trigonometric angle addition, and dot product are saying the same thing. My question ...
2
votes
2answers
70 views

Intuition of Greens Theorem in the plane

I'm trying to understand a special case of Greens Theorem. Let $V: \Omega \to \mathbb{R}^2$ be a $C^1$ vector field defined an open set $\Omega \subseteq \mathbb{R}^2$. Let $\gamma$ be a ...
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0answers
21 views

Finding a line integral by conservative field extension

Problem: Determine the values A and B for which the vector field \begin{align*} F = Ax \ln(z) \hat{i} + By^2 z \hat{j} + (\frac{x^2}{z} + y^3) \hat{k} \end{align*} is conservative. If $C$ is the ...
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1answer
60 views

Definition of divergence operator

There is the geometric definition of a divergence of a vertor field to be the following limit: How does this definition turns out to be the del operator dot the vector field in cartesian ...
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3answers
22 views

Verifying Vector Operation Identities

I'm having a hard time verifying these identities, anyone have any suggestions for any of them? For each Identity $F$ and $G$ denote vector fields, $\phi$ denotes a scalar field, and $R=xi+yj+zk$. ...
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30 views

Verifying the Divergence Theorem with Maple - concrete example

Let $\mathbf{F} = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k}$ be the flux outward across the boundary of the solid ellipsoid $x^2 + y^2 + 4(z-1)^2 = 4$. I now want to verify with Maple that the ...
2
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0answers
44 views

Application of Implicit Function Theorem in Munkres Analysis on Manifolds

I'm studying the Implicit Function Theorem and this is a problem from Munkres' Analysis on Manifolds. Let $F:\mathbb{R^2} \to \mathbb{R}$ be of class $C^2$, with $F(0,0)=0$ and ...
1
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1answer
18 views

Show function is a continuous function - Vector Calculus

I'm struggling to understand and how to approach this question, if you could give me a hint about how to answer it I would appreciate that. So here's the question: Show, by fixing the value of ...
0
votes
1answer
41 views

If the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are coplanar, then find the following determinant

If the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are coplanar, then find \begin{vmatrix} \vec{a} & \vec{b} & \vec{c} \\ \vec{a}\cdot\vec{a} & \vec{a}\cdot\vec{b} & \vec{a}\cdot\vec{c} ...
0
votes
1answer
57 views

Gram-Schmidt procedure on functions

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials. Here i what i understand If i have 2 functions, say ...
1
vote
1answer
43 views

Is the Mass flow rate (Mass flux) a scalar quantity?

Wikipedia states that mass flow rate is a scalar quantity, however Mass Flow Rate= Density x Cross Sectional Area x Velocity and velocity is a vector quantity, so this would imply Mass Flow Rate is ...
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0answers
39 views

Finding the image of multivariable functions

Let $f: \mathbb{R^2} \to \mathbb{R^2}$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy)$$ Let $A$ be the set consisting of all $(x,y)$ with $x \gt 0$. and $g: \mathbb{R^2} \to \mathbb{R^2}$ by ...
3
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0answers
19 views

Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
3
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0answers
66 views

Chain rule for the curl of a vector-valued function

I am looking for a vector expression for the curl of a composite vector-valued function. In other words $$ \nabla\times\mathbf{A}(\mathbf{B}) = ? $$ In indicial notation, this can be ...
2
votes
1answer
42 views

Evaluating a double integral over a hemisphere

Evaluate \begin{align*} \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{\hat{N}} \ dS, \end{align*} where $S$ is the hemisphere $x^2 + y^2 + z^2 = a^2, z \geq 0$ with outward normal, and $\mathbf{F} ...
0
votes
1answer
72 views

Why is $[\partial f/\partial x,2\partial f/\partial y,\partial f/\partial x]$ NOT a vector?

Gradient of a scalar function f is a vector. I just read a proof of why gradient is a vector. The proof follows from the fact that Directional derivative is not depended on choice of coordinates. ...
0
votes
1answer
17 views

scalar function's value - choice of coordinates

In a book it says that: "f is a scalar function. Hence its value at a point P depends on P but NOT on the particular choice of coordinates." I do not understand this statement. Its value depends on ...
0
votes
1answer
27 views

How to prove this vector identity using triple product?

Need to prove that (v⋅∇) v=(1/2)∇(v⋅v)+(∇×v)×v I could do it by applying the definitions directly, but triple product gives almost the right answer: (a×b)×c=-(c⋅b)a+(c⋅a)b In my case I get ...
0
votes
1answer
51 views

Gauss Divergence Theorem finding limits

Use Gauss Divergence Theorem to comput $$\int \int \limits_S F\cdot n dS$$ where $n$ is the outward normal for the following: $S$ is the surface of the unit sphere $\{(x,y,z): x^2+y^2+z^2=1\}$, $n$ ...
0
votes
1answer
31 views

For what value of $k$ is the vector field solenoidal

Problem: Let $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of a general point in $3$-space, and let $s= |\mathbf{r}|$ be the length of $\mathbf{r}$. For what value of the ...
0
votes
1answer
51 views

Gradient of a function defined on a surface

Let $V:R^{3}\rightarrow R$ be a differential function. Let $$A= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix}. ...
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0answers
76 views

using stokes thm on cylinder and sphere intersection

Use Stoke's theorem to solve the following integral (each time the curve is oriented counterclockwise when viewed from above): $$\int \limits_C xdx +(x-2yz)dy+(x^2+z)dz$$ where $C$ is the intersection ...
0
votes
1answer
52 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
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2answers
57 views

Showing a vector identity

Problem: If $\phi$ and $\psi$ are smooth scalar fields, show that \begin{align*} \nabla \times (\phi \nabla \psi) = -\nabla \times (\psi \nabla \phi ) = \nabla \phi \times \nabla \psi .\end{align*} ...
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2answers
241 views

Stokes' Theorem Example sphere

Been asked to use Stokes' theorem to solve the integral: $\int _C x dx + (x - 2yz)dy + (x^2 + z)dz $ where C is the intersection between $x^2+y^2+z^2=1$ and $x^2+y^2=x$ and the half space $z>0$. ...
0
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1answer
55 views

using stokes' theorem with curl zero

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

For a vector-valued function $F$, never zero with a continuous derivative always parallel to itself, prove that $F(t)=u(t)A$

I'm having trouble solving the following problem. A vector-valued function $F$, which is never zero and has a continuous derivative $F'(t)$ for all $t$, is always parallel to its derivative. Prove ...
2
votes
0answers
26 views

Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, show that$F''$ is orthogonal to $F'$

Given a nonzero vector $B$ and a vector-valued function $F$ such that $F(t)\cdot B=t$ for all $t$, and such that the angle between $F'(t)$ and $B$ is constant (independent of $t$). Prove that $F''(t)$ ...
0
votes
1answer
52 views

Integral over vector field

Can someone help me with this one: Let $X$ be a vector field on $R^{3}$ such that $X(x)=x$, for each $x\in S^{2}$. Calculate $$\int\limits_{\overline{B^3(1)}} \left(\frac{\partial X_1}{\partial x_1} ...
1
vote
1answer
42 views

Curl matrix operation

Consider a vector field $\underline{{f}}:\mathbb{R}^3\rightarrow \mathbb{R}^3$. We know that $\underline{\nabla}\cdot\underline{f} = tr(D\underline{f})$, $D\underline{f} = \begin{pmatrix} ...
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vote
2answers
39 views

Why is $\overrightarrow{OM}$ in that form?

We have the following: We have that $M$ is on the line segment $AB$. $\overrightarrow{OA}=\overrightarrow{a}$ $\overrightarrow{OB}=\overrightarrow{b}$ Could you explain to me why it stands ...