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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4 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 ...
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1answer
15 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 / ...
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0answers
6 views

Show the exceptation of a normalized vector [on hold]

Given $n$ $N \times 1$ vectors, $x_1$, $x_2$, ..., $x_n$, which are i.i.d complex Gaussian distributed with zero mean and variance one. Let $z=\sum_{i=1}^n x_i$. Please show that the exceptation of ...
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1answer
40 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 ...
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2answers
25 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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3answers
52 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
23 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
17 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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0answers
19 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 ...
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2answers
36 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 $\vec{a}$ , ...
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0answers
18 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
17 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? ...
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1answer
40 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
31 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 ...
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1answer
33 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 ...
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3answers
516 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 = ...
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3answers
77 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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2answers
38 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
21 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
42 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
24 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
24 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
13 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
19 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} ...
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2answers
26 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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3answers
42 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 ...
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1answer
20 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
13 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
29 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 ...
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2answers
46 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
17 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
46 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
21 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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0answers
19 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 ...
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0answers
16 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 ...
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1answer
17 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 ...
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1answer
38 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
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1answer
44 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 ...
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1answer
18 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
35 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 ...
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0answers
17 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
36 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
32 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} ...
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1answer
69 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. ...
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1answer
16 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
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1answer
25 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 ...
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1answer
27 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
23 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
37 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}. ...
1
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0answers
68 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 ...