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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29 views

Variational characterization of gradient?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. One way to define the gradient of $f$ is as the vector whose inner product with any other vector gives the directional derivative in ...
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1answer
31 views

How to find $\int_{S^2}f \cdot n \ \text{d}S$ if $f(x,y,z):=(x^3,y^3,z^3)^T$

With $\mathbb{S}^2$ being the unit sphere, how to find $$\int\limits_{\mathbb{S}^2} \vec{f} \cdot \vec{n} \ \text{d}S$$ if $\vec{f}(x,y,z):=(x^3,y^3,z^3)^T$? Apparently, we need to use Gauss. ...
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2answers
73 views

Tangent line of a convex function

Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$ $$ V(y) \geq ...
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2answers
25 views

Finding the magnitude of a vector product between two vectors?

Vector $\overrightarrow{A}$ has magnitude $11.0m$ and vector $\overrightarrow{B}$ has magnitude $16.0m$ . The scalar product $\overrightarrow{A}\bullet \overrightarrow{B}$ is $79.0m^2$. What is the ...
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1answer
77 views

Vector field on sphere

I want to find the gradient vector field and flows of the function $f=x^2+2y^2+3z^2$ on the sphere $S^2$, however I've not done this in a while so would appreciate a bit of help. I'd like to see the ...
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1answer
45 views

Physics Vector Problem - Airplane

Heres the question: A plane leaves the airport in Galisto and flies $140$km at $68.0^∘$ east of north and then changes direction to fly $255$km at $48.0^∘$ south of east, after which it makes an ...
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1answer
32 views

value of $|2a+5b+5c|$ if

If a, b, c are unit vectors satisfying $|a-b|^2+|b-c|^2+|c-a|^2=9$ then find the value of $|2a+5b+5c|$ Options are: A: $1$, B: $2$, C: $3$, D: $4$ considering $|2a+5b+5c|^2$ as $k$ then ...
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3answers
46 views

Problems with the integration law of Gauss.

I'm havin problems understanding the integration law by Gauss which states: $$\iiint\limits_{G} \operatorname{div}(\vec{w})\, dV = \iint\limits_{\partial G} \vec{w} \cdot \vec{n } \, dA$$ (I don't ...
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1answer
26 views

Show that the sequence $\Omega^0\Bbb{R}^2\ \longrightarrow\ \Omega^1\Bbb{R}^2\ \longrightarrow\ \Omega^2\Bbb{R}^2$ is exact.

I have been posed the following question, which I am unable to answer: Let $a_1,a_2\in\mathcal{C}^{\infty}(\Bbb{R}^2,\Bbb{R})$ be infinitely differentiable functions such that $\frac{\partial ...
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1answer
39 views

What does $\Delta$ mean in context of vector calculus?

I'm reading an article that has a formula for $\Delta \phi(x)$, where $\phi : \mathbf{R}^2 \rightarrow \mathbf{R}$ and $x \in \mathbf{R}^2$ and $\Delta \phi(x) : \mathbf{R}^2 \rightarrow \mathbf{R}$ ...
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1answer
189 views

Calculation of a curvilinear integral

Please help to calculate the following integral. TCalculate $$\int_\gamma \frac{x\,dx + y\,dy+z\,dz}{x^2+y^2+z^2}$$ where $\gamma$ is the way of class $\mathcal C^1$ which unites point on the ...
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0answers
55 views

Stokes Theorem (Application)

APPLICATIONS OF STOKE'S THEOREM STATEMENT-If a vector field R is irrotational then a line integral is independent of path. PROOF- Let $\nabla$ x $\vec A$=o in R consider the difference of two line ...
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1answer
26 views

How to determine the magnitude of a resultant vector?

I was able to determine a resultant vector based on the sum of two vectors and told to express them in vector units. Here is my answer that was correct: ...
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1answer
77 views

How must I understand concepts equations of physics?

I teach myself mathematics, but those days I wanted to learn about General relativity (not to pursue in it but only to have some background), perhaps because I am very curious to learn why exactly We ...
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1answer
26 views

Stokes's Theorem on a Curve of Intersection

Let $C$ be the intersection of $y+z=0$ and $x^2+y^2=a$ ($a>0$), oriented counterclockwise when viewed from above on the $z$-axis. Compute $$\int_C(xz+1)\text{ d}x+(yz+2x)\text{ d}y$$
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36 views

The 8 vectors to be made non-collinear

Consider the set of $8$ vectors $V=\{ai+bj+ck:a,b,c \in \{-1,1\}\}$. How can I choose three non-collinear vectors from $V$? My try: Let there be three vectors \begin{align*} ...
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1answer
20 views

Convert $(-\sqrt{2},1,0)$ in to cyclidrical and spherical coordinates

So for my answers I'm following the formulas given but I get stuck at finding $\theta$ because it equals $\pi + tan^{-1}(-\frac{1}{\sqrt{2}})$. Does that sound right?
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0answers
62 views

Proof of Gauss' Law of gravitation without reference to Newton?

Gauss' Law of gravity is: $$\bigtriangledown \cdot \mathbf{g}= 4\pi G\rho$$ This can be shown to be equivalent to Newton's Law of gravity via the divergence theorem. However, this does not really ...
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17 views

Derivative of a parametrized vector on a nonfixed basis

Suppose a curve defined by a vector parametrized through the variable $u$, and expressed on a non-fixed base, like the polar coordinates base. You derive it with respect to that parameter. What ...
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52 views

Integral curves in neighborhood of a minimum point of the potential

Let $X: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth vector field, and $f: \mathbb R^3 \rightarrow \mathbb R$ be its potential (i.e. $X= grad\ f$). When $f$ has a minimum point $p_{min}$ and all ...
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2answers
63 views

Multivariable Calculus - Calculating Derivative Matrix

I'm working with Munkres' Analysis on Manifolds. From chapter 2 (this isn't a homework question): Given $f: \mathbb R^2 \rightarrow \mathbb R^2 : f(r,\theta)=(r\cos(\theta),r\sin(\theta))$, ...
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How to solve $A \; x = 1/x$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$A \; x = 1./x, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $1./x$ denotes the ``element-wise inverse of the vector ...
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1answer
50 views

Find fourth coordinates given other three points

Find coordinates of $D$, given coordinates of $A,B,C$, torsion angle and angle between $BCD$. Is there any other way other than the torsion angle equation, $$n_1=⟨b_1\times b_2⟩ \;\text{ and }\; ...
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1answer
52 views

Simplifying the cross and dot product

Let A and B be arbitrary vectors. simplify $(a+2b)\cdot(2a-b)$ I did $(2a\cdot a)-(a\cdot b)+(2a\cdot b)-2(b\cdot b)$ $2[a^2]+ab-2[b^2]$ would this be correct 2.$(a+2b)\times(2a-b)$ ...
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1answer
94 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
2
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1answer
122 views

A silly mistake concerning spherical coordinates and unit vectors…

I'm quite comfortable with vector calculus in all sorts of coordinate systems, but for the love of me, I can't seem to figure out where did I go wrong in this simple derivation of the position vector ...
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1answer
31 views

Show, by finding a potential V(r) such that F = −∇V , that F is conservative

A particle at position r experiences a force: $$F=(-\frac{a}{r^2}+\frac{b}{r^3})\hat{r}$$ a and b are constants and $\hat{r}$ is the unit vector in a radial direction. I am told that I will need the ...
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3answers
69 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
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2answers
73 views

Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons ...
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1answer
60 views

Mapping Confusion -Implicit Function Theorem-

Here is the Implicit Function Theorem statement: "Let $g : R^k \times R^n \to R^n$ be a continously differentiable function s.t. $g(x_0, y_0) = c$ and $D_yg(x_0,y_0) : R^n \to R^n$ is an isomorphism. ...
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1answer
32 views

Volume integral of of two vector fields

The question is to evaluate $\iiint d^{3}r\vec{\nabla}\phi\centerdot\vec{G}$ when $\vec{\nabla}\centerdot\vec{G} = 0$ I started with $\iiint dxdydz (\frac{\partial\phi}{\partial ...
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2answers
68 views

Find the Unit Normal Vector - Calc III

For the curve given by: $r(t)= [\sin(t) - t\cos(t), \cos(t) + t\sin(t), 6t^2 + 2]$ Solve for the Unit Normal Vector N(t). I was successfully able to solve the Unit Tangent Vector $T(t)$ as ...
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2answers
1k views

Find the Vector Equation of a line perpendicular to the plane.

Question: Find the vector equation $r(t)$ for the line through the point $P = (-1, -5, 2)$ that is perpendicular to the plane $1 x - 5 y + 1 z = 1$. Use $t$ as your variable, $t = 0$ should ...
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1answer
38 views

Is this field conservative?

I'm looking for a potential of this field: $ F=\frac{f(r)}{r} (x,y,z) $ where $ r=\sqrt{x^2+y^2+z^2} $ and $ f $ is a $ C^1 $ function. Any help is appreciated. Thanks a lot!
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1answer
54 views

Why is this true: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\vec v \cdot (\nabla \cdot S)$?

I am doing fluid mechanics and I don't understand a particular step that is being used. It is the following step which I don't understand: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec ...
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2answers
53 views

Algebra-only vector product question

In the case when a=ai, b=bj and c=ck, where a, b, and c are positive scalar constants, determine the equation of the plane (which contains a, b, and c) in the form r.n=d, where the components of n and ...
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40 views

What does the ~ symbol, placed above a vector, mean

On the wikipedia page for the Poynting Vector, under the section 'time averaged Poynting vector', the ~ symbol is used above some of the vectors. What does this mean? Thank you.
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0answers
95 views

A nabla operator identity in ortho-normal coordinate systems

Let $u_1,~u_2,~u_3$ be components of the position vector $\vec{u}$: $$\vec{u}=u_1 \vec{e}_{u_1}+u_2 \vec{e}_{u_2}+u_3 \vec{e}_{u_3}$$ in an ortho-normal coordinate system (not necessarily Cartesian) ...
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0answers
19 views

Irrotational implies path independent

I wanted to prove that if $F$ is differentiable and irrotational then it's path independent. The plan of actions is straightforward: let $C_1$ and $C_2$ be two curves, then $C$ is a closed curve then ...
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1answer
91 views

What if $\operatorname{div}f=0$?

Say, we have a function $f\in C^1(\mathbb R^2, \mathbb R^2)$ such that $\operatorname{div}f=0$. According to the divergence theorem the flux through the boundary surface of any solid region equals ...
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1answer
122 views

Finding the sphere surface area using the divergence theorem and sphere volume

The divergence theorem allows us to go between surface and volume (in some sense), so a natural example would be to compute the surface area of the unit sphere $U$ assuming we know the sphere volume. ...
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3answers
62 views

Relating geometric and Algebraic Definitions of the dot product

I am about to enter the class Engineering Physics II. Alas, much of my mastery of vector manipulation is predicated on something I don't understand that must be taken as an assumption for me to ...
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1answer
40 views

If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ …

[2D - vector application] If $r = \langle x,y\rangle$, $r_1 = \langle x_1, y_1\rangle$, and $r_2 = \langle x_2,y_2\rangle$, describe the set of all points $(x,y)$ such that $|r - r_1| + |r - r_2| = ...
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2answers
81 views

Del operator ($\nabla$) in spherical co-ordinate system

I am teaching myself about vector fields and came across the following question: Is the following force field $\vec{F}$ conservative, where $\vec{F}(r,\theta,\varphi)$ is defined by: ...
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1answer
128 views

Slick proof of Gauss' theorem

Below is a very concise proof of Gauss's theorem (from the book Vector and Tensor Analysis with Applications by Borisenko, Tarapov). Unfortunately, I'm having trouble understanding it, despite staring ...
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2answers
53 views

zero of vector field with index 0

I'm currently studying vector fields on surfaces in the $\mathbb R^3$ and I currently I am doing some reading on the index of zeros of vector fields, which got me wondering: Is it possible to find a ...
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1answer
24 views

zeroes of vector fields on surfaces

I know that for compact (smooth) surfaces in $\mathbb R^3$ the 2-torus is the only surface that has vector fields with no zeroes. What happens if we take the compactness of the surface away? Does this ...
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1answer
15 views

Angles in a Triangle and Vectors

Given are two forces $P_1$ and $P_2$ which both act upon a particle point P, and the angle between the two forces is 40 degrees. Furthermore $P_1$ has an absolute value of 20 and $P_2$ of 12. What is ...
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1answer
116 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...