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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Evaluation of an Integral in Vector Analysis

I'm trying to calculate an individual probability $P(\hat{a})$ from a joint probability $P(\hat{a},\hat{b})$ in a physics application, where $\hat{a},\hat{b}$ are unit vectors. I need to evaluate the ...
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1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
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1answer
30 views

Field lines of vector field

Okey if $\phi(x,y)=\ln(x^2+y^2), (x,y) \neq (0,0)$. Find the field lines for $\mathbf{G}=\nabla \phi$. So $\mathbf{G}=\frac{2x}{x^2+y^2}\mathbf{i}+\frac{2y}{x^2+y^2}\mathbf{j}$ right? To find the ...
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Projecting spherical components of a variable point to the unit vector of a fixed point

Consider the following vector function in spherical coordinates: $\mathbf{v} (r, \theta, \phi) = V_{\phi} (r, \theta, \phi) \mathbf{a}_{\phi} = A \delta(r - k) \displaystyle \delta \left( \theta - ...
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1answer
30 views

Basic line integral

Let $C$ be curve along surfaces $z=\ln(1+x)$ and $y=x$ from $(0,0,0)$ to $(1,1,\ln(2))$. Calculate the work done by vector field $$\mathbf{F}(x,y,z)=(2x\sin(\pi y)-e^z)\mathbf{i}+(\pi x^2 \cos (\pi ...
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1answer
20 views

Understanding Conservative and Curl

There are several things I need to clarify on Curl. 1) Is the conservativeness of a gradient field only applicable for a Closed curve? If the field is gradient and if c (curve) is not closed then ...
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1answer
34 views

find flux,using Cartesian and spherical coordinates

Find the flux of the vector field $\overrightarrow{F}=-y \hat{i}+ x \hat{j}$ of the surface that consists of the first octant of the sphere $x^2+y^2+z^2=a^2(x,y,z \geq 0).$ Using the Cartesian ...
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36 views

an iterated integral question

This iterated integral is proving harder than I thought. Evaluate by reversing the order of integration: $$ \int_{0}^{1}\left(\int_{y=x}^{\sqrt{x}}\frac{\sin y}{y}dy\right)dx $$
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39 views

Determine if the vector field $\overrightarrow{F}$ is conservative or not.

Determine if the vector field $\overrightarrow{F}=y\hat{i}+(x+z)\hat{j}-y\hat{k}$ is conservative or not. The vector field $\overrightarrow{F}=M\hat{i}+N\hat{j}+P\hat{k}$ is conservative if ...
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9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
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21 views

Finding a vector potential, i.e. given $\vec{A}$, how to find $\vec{B}$ s.t. $\vec{A} = \nabla \times \vec{B}$?

I understand that this might not be unique, but is there a (relatively) painless way to generate such a 'vector potential', so for a given field $\vec{A}$, a new field $\vec{B}$ which satifies: ...
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A method of calculation coordinates in order to implement it to a code language!

lets say that we have three points A(xa,ya,za), B(xb,yb,zv), C(xc,yc,zc) with known coordinates in 3d space. Is there a method to calculate the coordinates (x,y,z) of another point D for which the ...
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1answer
17 views

Need help proving second partial derivative equation.

I need help doing a Midterm Practice Question. Attempt: I tried finding each individual partial derivative and adding them together initially. I ended up getting something extremely complicated ...
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1answer
25 views

Spread out field lines and divergence

It can be shown by calculation of divergence of a field like $(\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}})$ that it's divergence is positive. But I can't understand the geometrical essence ...
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Integrating a vector field, not uniquely defined in which sense?

I'm trying to complete an exercise in vectorial calculus, integrating the following vector field in the sense finding (one of) the function it derives from by taking the gradient: $$ \vec{v} = ...
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1answer
40 views

Use the Stokes' Theorem to find the work of the vector field $ \overrightarrow{F}$

I have the following exercise: "Use the Stokes' Theorem to find the work of the vector field $ \overrightarrow{F}=x^2 \hat{i}+2x \hat{j}+z^2\hat{k}$ along the anti-counterclockwise oriented area of ...
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25 views

Question on vector fields

Which ones are vector fields? (I checked my answers) Temperature of room at given point The gravitation that object with mass creates (x) The density of an object at given point Function $f: ...
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1answer
21 views

domain using level curves?

Let $f(x,y)=4x^2-y^2$. The problem is to determine the range of this function using the idea of level curves. So one sets $f(x,y)=D$, where $D\in\mathbb{R}$. What values can $D$ take?
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31 views

How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...
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134 views

A generalization of the mean value theorem?

Let $U \subset \mathbb{R}^d$ be open and path-connected. Let $f: U \to \mathbb{R}^m $ be differentiable on $U$ and suppose there exists a real $M$ such that $|| D_f(x) || \leq M $ for all $x \in U$. ...
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1answer
28 views

surface that is created by the intersection of paraboloid and plane

Find the surface that is created by the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$. $$x^2+y^2-z=0 \Rightarrow x^2+y^2=z$$ $$z=2$$ Is the intesection: $x^2+y^2=2$? So is the asked ...
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1answer
47 views

Why generalize vector calculus with $k$-forms instead of $k$-vectors?

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of ...
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1answer
37 views

Use the Green's Theorem to calculate the work and the flux

Use the Green's Theorem to calculate the work and the flux for the closed anti-clockwise direction that consists of the square which is determined by the lines $x=0$, $x=1$, $y=0$ and $y=1$ if ...
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1answer
43 views

Difference between Vector Functions and Vector Field

I understand that a vector function is a function that has a domain $\mathbb{R}^n$ and range on $\mathbb{R}^m$ so it takes vectors and gives vectors right? So what is a vector field?And how can I ...
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80 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
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2answers
58 views

Verify Green's Theorem-Calculate $\int \int_R{ \nabla \times \overrightarrow{F} \cdot \hat{n}}dA$

Given that $$\vec{F}=-x^2y \hat{i}+x y^2\hat{j}$$ $$C:r=a \cos{t}\hat{\imath}+a \sin{t} \hat{\jmath}, 0 \leq t \leq 2 \pi \text{ and } R: x^2+y^2 \leq a^2$$ I have to calculate $\iint_R{ \nabla \times ...
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3answers
73 views

How to deduce this formula using differential forms?

There's a formula from vector calculus that seems terrible to deduce. This formula is: $$\nabla\times (A\times B)=(B\cdot\nabla )A-(A\cdot \nabla)B+A (\nabla\cdot B)-B(\nabla\cdot A)$$ Deducing it ...
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1answer
38 views

plotting cones and space curves using Maple

A curve $\cal{C}$ is given by the parametric equation $$ {\bf r}(t) = (e^t\cos t, e^t\sin t, e^t-1)\;,\quad 0\leq t\leq 2\pi\;. $$ Using Maple, I would like to sketch the curve $\cal{C}$ and the cone ...
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1answer
26 views

Fractional change in volume from scale-factor

I was given the following question which I am unable to get a seemingly correct answer from: A body expands linearly by a factor $\alpha$ due to an increase in temperature. Because of the ...
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1answer
72 views

How can I find $ \int \int_R{ \bigtriangledown \times \overrightarrow{F} \cdot \hat{n}}dA $ ??

$$\overrightarrow{F}=-y\hat{i}+x\hat{j}$$ $$C:r=a \cos{t}\hat{i}+a \sin{t}\hat{j}, o \leq t \leq 2 \pi$$ $$R:x^2+y^2 \leq a^2 $$ Green's Theorem: $$\oint_C{F}dr=\int \int_R{\bigtriangledown \times ...
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1answer
33 views

A particle has the following path: $\vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j}$

A particle has the following path: $$\vec{r}(t)=t^2\hat{i}+(t^3-4t)\hat{j}$$At $t_0=2$ the particle fudges (leaves by the tangent). What is the position of the particle at $t=3$? The particle has ...
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0answers
29 views

Integration of Vector Difference, Coordinate Transformation

I have to calculate the following integral: $$I(\textbf{r})=\int d\textbf{r'} \frac{1}{|\textbf{r'}-\textbf{r}|}\cdot e^{-2r'-\lambda |\textbf{r'}-\textbf{r}|}$$ I thought of rewriting ...
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1answer
29 views

orthogonally intersecting families

Take two families of circles $$ (x-c_1)^2+y^2=c_1^2\qquad\mbox{and}\qquad x^2+(y-c_2)^2=c_2^2\;, $$ where $c_1$ and $c_2$ are positive constants. Use the gradient to show that these two families ...
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1answer
41 views

Green's Theorem: Integrate the $(\bigtriangledown \times F) \cdot k$

$$F=(xy^2, y+x)$$ Integrate the $(\bigtriangledown \times F) \cdot k$ over the region of the first eightant that is bounded of the curves $y=x^2$ and $y=x$. $$$$ Green's Theorem: $$\int \int_R ...
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0answers
12 views

Closed ans Exact Chains in Projective Space

I don't know why this is obvious, but... ...how can be show that $\forall n \geq 1$, exists, in $\mathbb{P}^n$(the projective space), chains that are closed but not exact? and how can be show that ...
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1answer
19 views

Integration by parts, gardient & laplacian

I saw the following statement: If D is a geometric manifold without boundary in $R^{d}$, then $ \int_{D} \nabla (f) . \nabla(g) \mathrm{d}x= - \int_{D} f . \Delta(g) \mathrm{d}x$ for $f, g \in ...
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3answers
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Question about potential function of a vector field

Lets say we have conservative vector field $\mathbf{F}(x,y,z)=y\sin{z}\mathbf{i}+x\sin{z}\mathbf{j}+xy\cos{z}\mathbf{k}$ I found it's potential function like this $$\begin{cases} ...
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Some questions about the proof of the divergence theorem

I have some questions about the following proof of the divergence theorem: $$z_1=f_1(x,y), (x,y) \in \mathbb{R}$$ $$z_2=f_2(x,y), (x,y) \in \mathbb{R}$$ $$\overrightarrow{F}=M \hat{i}+N \hat{j}+P ...
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Conservative vector field in polar coordinates

Find the potential function of a conservative vector field $\mathbf{F}(r,\phi)=r\sin(2\phi)\mathbf{\hat{r}}+r\cos(2\phi)\mathbf{\hat{\phi}}$ Does my solution seem right: So if ...
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3answers
58 views

Explain Normalization in Layman's term

Can someone explain me what is Normalization in Layman's term ? If we have a vector a, we normalize it by dividing it by |a|. That is $$\frac {a}{|a|} $$ Why we need normalization?
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66 views

A line integral

If $\mathbf{B}(\mathbf{x})=\rho^{-1}\mathbf{e}_{\phi}$ in cylindrical polars, find: $$\int_{C}\mathbf{B}\cdot\mathrm{d}\mathbf{x}$$ where $C$ is the circle $z=0,\rho=1,\;0\leq \phi\leq ...
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3answers
53 views

Vector valued function in $\mathbb{R}^2$ with non-finite arc length

Find a curve $\mathcal{C}$ : x = g(t) , $a \leq t \leq b$ in $\mathbb{R}^2$, where g $\in \mathcal{C}[a,b]$ such that $\mathcal{C}$ does NOT have finite arc length
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50 views

Surface integral

Problem statement $$ \mbox{Calculate the surface integral}\quad \int_{Y}\ y\,\sqrt{z\,}\,\sqrt{4x^{2} + 4y^{2} + 1\,}\,\,{\rm d}S $$ where $Y$ is the surface $\left\{\left(x,y,z\right)\ \ni\ ...
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1answer
25 views

Line Integral using Green's theorem

Problem statement: Calculate $\int_{\gamma }(3e^{(y-3x)^{2}}-y)\mathrm{dx}+(-e^{(y-3x)^{2}}+2x)\mathrm{dy}$ where $\gamma$ is the curve $y=x^2$ from $(0,0)$ to $(3,9)$. Progress First idea was to ...
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Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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41 views

Calculating grad/curl/div of a vector?

I'm trying to do some practice for an electromagnetism course, and am trying to calculate the grad, curl and div of a vector $A = (2xy, 3zx, yx^2)$ I know: Div = $\nabla . A$ Curl = $\nabla \times ...
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1answer
38 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
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1answer
20 views

Solve for the tangent plane using the gradient

I am having a hard time finishing this problem up: Consider the surface $4 x^{2} + 9 y^{2} + 4 z^{2} = 17$ and the point $P = \left( 1, 1, 1 \right)$ on this surface. A) Find the outward unit ...
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2answers
173 views

Directional Derivatives using Polar Coordinates

I am having a hard time with this problem on my homework assignment. Here is the problem, and i will show my work below: If $f( x, y) = -2 x^{2} + 3 y^{2}$, find the value of the directional ...
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1answer
26 views

Does this limit of arc length exist?

We have a parametrized curve $\gamma: \mathbb{R} \rightarrow \mathbb{R^2}$ given by $\gamma (t) = \langle e^t\cos (t), e^t\sin(t)\rangle$. I want to compute the arc-length of this curve on $[a,b]$ in ...