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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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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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
65 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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34 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
56 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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25 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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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)
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28 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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84 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 ...
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1answer
17 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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764 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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15 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
47 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
27 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
29 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
47 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
34 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
50 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
34 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
31 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
16 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
54 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
60 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
28 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
33 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
29 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 ...
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1answer
21 views

Calculate line integral without integrating

Without carrying out any integration, show that the line integral $$\int_{C} \nabla\phi\cdot \mathbf{dr} = 3$$ where $C$ is any smooth curve joining $(-1, 3, 9)$ to $(1, 6,-4)$ and $\phi = xyz$. I ...
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1answer
50 views

surface integrals parametrising

Find a parameterisation and compute $r_{\alpha},r_{\beta},r_{\alpha}$ x $r_{\beta}$ and the tangent plane at the point mentioned of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ and the ...
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1answer
41 views

If divergence is zero, is it necessarily a curl?

The divergence of the curl of a vector is zero. But, Any vector whose divergence is zero can be the curl of a vector field ?
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1answer
30 views

Find $F_u(1, 1)$ and $F_{u,v}(1, 1)$.

Need help on this.. Suppose that $F(u, v) = f(x(u, v), y(u, v))$, where $f$ is a function satisfying \begin{cases}f(1, 2) = 3\\f_x(1, 2) = 1\\f_y(1, 2) = −2\\f_{x,x}(1, 2) = 3\\f_{x,y}(1, 2) = ...
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1answer
36 views

Show that $xg_x(x, y) + yg_y(x, y) = 0$.

Need help with this. Suppose that $G(u, v)$ is a differentiable function of two variables and that $g(x, y) = G(x/y , y/x)$. Show that $xg_x(x, y) + yg_y(x, y) = 0$. Where $g_x(x,y)$ and $g_y(x,y)$ ...
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0answers
40 views

What is meaning of $FFT(\vec{E}(x,y ))$

What is the meaning and how one takes fourier transformation of vector that has spatial distrubution. Let say electric field (with transfer x, y distibution) with direction $$FFT(\vec{E}(x,y ))$$ ...
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25 views

Projection of Vectors v on w

Given $v = [3, -6, 2]$ and $w = [-1, 6, 5]$, find; $v \downarrow w$ $w \downarrow v$ What does the magnitude of $w \downarrow v$ depend on? What does the direction of $w \downarrow v$ depend on? ...
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1answer
18 views

$|V|=?$ if $2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j}))=2\vec{i}+\vec{k} $

I am given $$2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j})=2\vec{i}+\vec{k} $$ and I need to find $$|V|=?$$ , I took dot product of equation with $\vec{V}$ but got stuck in $$2V^2 = ...
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2answers
66 views

$\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane

We want to show that if $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane, then there are $A, B, C$ not all $0$ such that $A \vec a+B \vec b+C \vec c=\vec 0$. $$$$ ...
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1answer
22 views

Two parallel planes

When two planes have the same perpendicular vector $\overrightarrow{v}$, then they are parallel, right?? We have that the two planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ are parallel, since they ...
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1answer
70 views

Triangle in space

Using vector notation describe the triangle ( in space ) that has as vertices the origin and the endpoints of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. $$$$ The solution is the ...
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1answer
51 views

Velocity vector

We suppose that a ship, that is at the position $(1, 0)$ of a nautical map (with the North at the positive direction $y$) and it "sees" a rock at the position $(2, 4)$, is directed to North and is ...
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2answers
46 views

Angular velocity

The angular velocity $\omega$ of rotation of a rigid body has the direction of the rotaion axis and magintude equal to the rotation rate in rad per second. The orientation of $\omega$ is determined by ...
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1answer
57 views

Torque of a force

The torque $M$ of a force $\overrightarrow{F}$ as for the point $O$ is defined as the product of the magnitude of the force $\overrightarrow{F}$ and the perpendicular distance of the point $O$ and the ...
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2answers
75 views

Do these vector fields span an integrable distribution?

Let $X_i$, $1 \le 1 \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that ...
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1answer
38 views

Show that it is the volume

A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$. Let $\overrightarrow{n}$ an unit vector perpendicular to the plane. Show that $\overrightarrow{v} \cdot ...
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0answers
54 views

proving the laplacian of a vector in cylindrical coordnates

I am proving the following identity for the laplacian of a vector $\vec{v}=<v_r,v_\theta,v_z>$ in cylindrical coordinates: $$\nabla^2 \vec{v}=\left( \frac{\partial^2 v_r}{\partial ...
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0answers
79 views

Derivative of angle between two vectors singularity!

I have been battling a problem of needing to know the derivative of the angle between two vectors, the vectors possibly being parallel at some points in time. I started off with: $$\bf A \dot \bf B = ...
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2answers
32 views

finding potential f

Compute $f$ so that $F=\nabla f$ for $$F(x,y,z)=(yz+x-y, xz-x+z, xy+y-z)$$ I need to see the method of finding this that is not using the theorem about the star shaped set. The method that is like ...
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2answers
57 views

Poincaré–Bendixson theorem

Does someone know a good reference for a proof of the Poincaré–Bendixson theorem using the language of vector fields?
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2answers
20 views

Is the work integral decomposable?

Work is defined as $W = \int_{\gamma} \vec F \cdot d\vec l$ which I think means $W = \int (F_x, F_y, F_z) \cdot (dx, dy, dz)$. So by the linearity of the integral, could we always decompose work into ...
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4answers
107 views

Find the equation of the plane knowing that it passes through 3 points

I have to find the equation of the plane that passes through $(0, 0, 0), (4, 0, -2), (0, 8, -6)$. I have done the following: The equation of the plane is of the form $$ax+by+cz+d=0$$ Since the ...