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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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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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 ...
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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 ...
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1answer
35 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
71 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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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 ...
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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 ...
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1answer
39 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 $\{(x,y,z): x^2+y^2+z^2=1\}$, $n$ ...
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1answer
25 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 ...
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1answer
43 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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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 ...
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1answer
44 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
53 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
219 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 $z>0$. ...
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1answer
49 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 ...
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1answer
23 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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21 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 $F''(t)$ ...
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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} ...
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1answer
39 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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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 ...
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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 ...
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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 ...
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1answer
19 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 ...
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27 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?
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2answers
53 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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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: ...
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1answer
38 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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20 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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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)
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27 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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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 ...
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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 ...
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17 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 ...
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609 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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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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37 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 ...
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3answers
22 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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1answer
26 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 ...
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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 ...
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1answer
44 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 ...
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1answer
29 views

Product of projections of equispaced rotating vector

When equal and equi-spaced forces are summed on y-axis what is vector sum? How do we derive the formula $$ \sum_{k=1}^{n-1}\sin\frac{\pi k}{n} = \cot \frac{\pi}{2 n} $$ ( Formula given by ...
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2answers
45 views

defining vectors on a manifold

I've recently started studying differential geometry and I'm a bit unsure on the notion of a tangent vector on a manifold. Is the point that we can no longer thing of a vector as an arrow (a straight ...
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1answer
33 views

A question on defining tangent vectors on a manifold and their “defining” theorem

In the appendix of Lovelock's book "Tensors, Differential Forms and Variational Principles" they give a proof of a theorem fundamental to the notion of a tangent vector on a manifold: Part of the ...
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1answer
30 views

Find potential function for the vector field $\vec F(x)=\left \| x \right \|^px$

Define a vector field $\vec F$ in $\Bbb{R}^n \setminus 0$ by $\vec F(x)=\left \| x \right \|^px$, where $p$ is a real constant. How to find a potential function for $\vec F$? Shall I just directly ...
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1answer
15 views

show that $c′(5)$ is orthogonal to $\nabla f(1,4,2).$

I need some help here. Let $f(x, y, z)$ be a differentiable function and suppose that $c(t)$ is a path which lies on the surface $f(x, y, z) = 17.$ If $c(5) = <1, 4, 2>$ show that $c′(5)$ is ...
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2answers
44 views

When are there not min/max values of a function subject to a constraint?

How do I know if there are no extreme values of a function subject to a constraint? For example, if $f(x,y,z)=xy+3xz+2yz$ subject to the constraint $5x+9y+z=10$. Why does it not have min/man values?
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2answers
58 views

Expand $(\vec{A}\times \nabla)\times \vec{B}$ using tensorial notation

I was given a task to prove $$(\vec{A}\times \nabla)\times \vec{B} = (\vec A \cdot \nabla)\vec B + \vec A \times \operatorname{rot} \vec B - \vec A \operatorname{div} B$$ using tensorial notation ...
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1answer
27 views

Finding the surface integral of a vector field

How many cubic metres of fluid cross the upper hemisphere $x^2+y^2+z^2=1$, $z\ge0$ per second if the velocity of the flow is $\mathbf{u}=\mathbf{i}+x\mathbf{j}+z\mathbf{k}$ metres per second. So I ...
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3answers
30 views

Parametrising a surface for a surface integral

How many cubic metres of fluid cross the upper hemisphere $x^{2}+y^{2}+z^{2}=1$, $z\ge0$ per second if the velocity of the flow is $\mathbf{u} = \mathbf{i}+x\mathbf{j}+z\mathbf{k}$ metres per second. ...
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1answer
28 views

parametrising a surface

I want to parametrise the surface $x^{2}+y^{2}=36$ to then calculate a surface integral however I'm not really sure how to parametrise this. Can we use $$\mathbf{r}(t) = (6\cos{t}, 6\sin{t}, 0).$$ I ...