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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2answers
48 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 ...
0
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1answer
47 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 ...
0
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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
21 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 ...
0
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1answer
39 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
46 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
21 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 ...
1
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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 ...
3
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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
38 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
33 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
70 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
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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 ...
0
votes
1answer
29 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
40 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
vote
0answers
70 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
42 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, ...
0
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2answers
48 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*} ...
0
votes
2answers
181 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
votes
1answer
46 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
18 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 ...
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0answers
19 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
51 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
34 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} ...
1
vote
2answers
36 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 ...
1
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0answers
29 views

Definition of 2D curl in textbook or paper

The curl in 2D is commenly written as $\text{curl}_{2D}(u) = \frac{\partial u_2}{\partial x_1} - \frac{\partial u_1}{\partial x_2}$. It is often described as thinking of 3D field with the thrid ...
0
votes
2answers
27 views

Given two orthogonal vectors $A$, $B$, in $\mathbb R^3$ each of length 1. Let $P$ be a vector satisfying the equation $P\times B=A-P$.

Given two orthogonal vectors $A$, $B$, in $\mathbb R^3$ each of length 1. Let $P$ be a vector satisfying the equation $P\times B=A-P$. Prove each of the following statements. (a) $P$ is orthogonal to ...
0
votes
1answer
18 views

A problem on cross products to find angles and length

Given two linearly independent vectors $A$ and $B$ in $\mathbb R^3$. Let $C=(B\times A)-B$ (a) Prove that $A$ is orthogonal to $B+C$. (b) Prove that the angle $\theta$ between $B$ and $C$ satisfies ...
1
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0answers
26 views

Find surface area for $f(x,y,z)=e^{−z}$, over $x^2+y^2=9, 0≤z≤3$

I am having trouble parametrizing $f$, since $z$ does not seem to be related to $x$ and $y$ in any way. Any hints?
0
votes
2answers
50 views

Introducing new indices with tensor/index notation

I understand where the $k$ comes from in $\varepsilon_{klm}$, however why do we need to introduce $l,m$ rather than continuing with $i,j$, i.e $\varepsilon_{kij}b_ic_j$
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0answers
28 views

Find the direction from which the projected area of a loop is maximal

How do I find the direction from which the projected area of a loop is maximum? Should I try to use intuition or is there a simple mathematical way to find it? The problem given was the following: ...
1
vote
1answer
33 views

Multiplying Gradients in Vector Calculus

What happens when you multiply two gradients of two scalar fields together? So: $$ \vec{\nabla}A\cdot\vec{\nabla}B $$ Using Einstein summation convention I get: $$ ...
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0answers
17 views

Finding the flux integral over a surface

Consider the vertical vector field $\mathbf{u}=-\mathbf{k}$ representing a constant downward flow of rain on a sloped roof $S:z = 4-2x-y$ where $x, y, z\ge0$. Find the velocity flux in the negative ...
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0answers
20 views

Find points of nonintersecting lines

Given two nonintersecting lines $x/2=y=(z-1)/3$ and $x/3=y=z$ find points $P$ on the first line and point $Q$ on the second line so that $PQ$ is perpendicular to both lines.(Vector algebra)
0
votes
1answer
26 views

Where does this result come from?

I'm sorry about the non-specific title, I wasn't sure where this question would fit in... I'm reading through a few notes for my PDE course, and I'm struggling to see where the following result comes ...
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0answers
73 views

What are these integrals used for?

In a primer on differential geometry, it was mentioned that each of these are different types of line integrals: $\int_C f(\vec r) |d\vec r|$, $\int_C \vec F |d\vec r|$, $\int_C f(\vec r)d\vec r$, and ...
0
votes
1answer
14 views

Norm of vector equals norm of it's basis representation

I will try to represent my question by example. There is a vector $a \in R^d$, basis $b$ spans $R^d$, so vector $a=\sum_{i=1}^{d}c_i b_i$. Whether $\left \| a \right \| = \left \| c \right \|$? If ...
0
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0answers
16 views

Compute integrals involving a local basis

A local basis is one whose vectors change direction and/or magnitude from point to point. Consider for example the 3D spherical coordinates: $ r,\theta,\phi$ that are the radial distance, azimuthal ...
5
votes
8answers
587 views

Distance of two lines in $\mathbb{R}^3$

Using vector methods show that the distance between two non parallel lines $l_1$ and $l_2$ is given by $$d=\frac{|(\overrightarrow{v}_1 - \overrightarrow{v}_2) \cdot (\overrightarrow{ a}_1 \times ...
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0answers
12 views

How does this bundle of gradients and a divergence equal these gradients and Hessians?

In words it may be vague, but my question is: why does $$ \nabla \cdot (\nabla f \cdot \nabla g) = Hess(f) \cdot \nabla g + Hess(g) \cdot \nabla f) $$ In d dimensions? Edit: After the hint, I was ...
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0answers
33 views

A Formal proof of Green Theorem

I want to go through the formal proof of Green theorem on a regular, simple and closed curve oriented counterclockwise and the vector space $F$ is a continuously differentiable vector field such that ...
0
votes
3answers
19 views

When is a vector field the curl of another vector field?

Under what conditions does a given vector field $\bf X$ on some open subset $U \subseteq \mathbb{R}^3$ satisfy ${\bf X} = \text{curl } {\bf Y}$ for some vector field $\bf Y$ on $U$?
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vote
1answer
22 views

Parametrizing intersection of a plane and surface

I'm working on… Parametrize the curve which is the intersection of the plane $2x+4y+z=4$ with the surface $z=x^2+y^2$. I tried eliminating $z$ by plugging it into the first equation and also ...
1
vote
1answer
36 views

finding bounds of parametric variables

Compute the area of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ You can get it in the form $x^2+(y-1)^2=(z+1)^2$ I parametrised it as $r(u,v)=(u\cos v, u\sin v+1, u-1)$. I know that ...
1
vote
1answer
39 views

Does every shape have zero volume?

Consider the digram below: the red line ($c$) enclosing an area on the XY plane lies in the yz plane and the blue line is a surface with this line as its boundry curve. Let's say we are trying to ...