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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26 views

Show that $\iint_S (n\times \nabla)f\, dS=\int_C tf\, ds$ and $\iint_S (n\times \nabla)\times f \,dS=\int_C t\times F\, ds$

If $S$ be a closed region lying on a surface and bounded by the curve $C$ and $n$ be the unit positive normal vector to $S$, and $f$ and $F$ be two fields with continuous fiest derivatives in $S$, ...
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2answers
70 views

Prove that $\int_S n\times r dS=0$

If $r$ be the position vector of a point on a closed surface $S$ and $n$ be the unit normal (outward) vector to $S$, then prove that $$\int_S n\times r\,dS=0$$ Attempt: $r=xi+yj+zk$, ...
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2answers
38 views

What does $\text{div} (A \text{ grad }b)$ mean?

I often see this term in my Applied Mathematics course. If $\text{grad }b= \frac{\partial b}{\partial x}\hat{i}+\frac{\partial b}{\partial y}\hat{j}+\frac{\partial b}{\partial z}\hat{k}$, then would ...
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0answers
14 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
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1answer
29 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
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0answers
33 views

a problem on stokes' theorem

the problem is as following Use stokes theorem to evaluate $\oint F.dr$ where, F = (-2Z) i + (X) j - (X) k , C is the ellipse $X^2 + Y^2 = 1 $ and $ Z = Y + 1 $ my solution is to get $curl F $ ...
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1answer
18 views

Calculating a line integral around a closed curve.

Let $u_0$ be a fixed vector, and let $b=u_0\times r$, where $r$ is the position vector $x\hat{i}+y\hat{j}+z\hat{k}$. What is $\int_C b.\hat{T}ds$, where $C$ is a closed curve? Assuming ...
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0answers
24 views

The gradient of the magnitude of the cross product of a constant vector and the position vector

If $ \underline c$ is a constant vector and $\underline r$ is the position vector. How can I show that $ \lvert \underline c \land \underline r \rvert grad \lvert \underline c \land \underline r ...
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2answers
16 views

Prove that, for $n, l \in \mathbb{N}$ the identity $\vec\nabla \times (f^n \vec\nabla(f^l)) = \textbf{ $\vec 0$} $

a) Let $f$ and $g$ be two smooth scalar fields. Prove the following identity: \begin{equation} \vec\nabla \times (f \vec\nabla g) + \vec\nabla \times (g \vec\nabla f) = \textbf{$\vec 0$} ...
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1answer
39 views

Intuitive explanation of the potential function of a vector field

Suppose I have some vector field $$\vec{f}(x,y)=\begin{pmatrix}A(x,y)\\B(x,y)\end{pmatrix}$$ then the potential function (if the field is conservative) can be found by integrating $A$ with respect ...
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1answer
32 views

Surface Integral over a Vector Field question

pretty basic question but I can't seem to work it out: Question: Let S be the triangle with vertices $\mathbf{a}=(1,2,3),\mathbf{b}=(1,1,1),\mathbf{c}=(3,1,2)$ with unit normal chosen such that the ...
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2answers
52 views

Linearity of Multilinear Maps

If $f: \mathbb{R^n} \rightarrow \mathbb{R^m}$, with $n>1$, is a multilinear map, is $f$ linear? I think $f$ is only linear for the special case that the range of $f$ consists of a single element, ...
2
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1answer
16 views

Orthogonal decompostion for $u^´(t)$

$u(t) $ is differentiable vector function in $\mathbb{R}^3$ on $[a,b]$ and $u(t) \neq 0$ for all t. $u^´(t)$ is the derivative of $u(t)$ and is orthogonal for $t \in (a,b)$ for all t $\implies$ ...
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3answers
82 views

Can you multiply two vectors $v$ and $u$ using multiplication instead of dot or cross products

On Wikipedia page on vector multiplication, they gave several ways to "multiply" two vectors, the most well known being the dot product and the cross product I am curious whether we can multiply two ...
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1answer
26 views

Finding Eigenvectors when we have lots of zeroes

\begin{array}{cc} 0 & 0 \\ 0 & 8 \\ \end{array} I have $\lambda_1=8$ and $\lambda_2=0$ but cannot find $V_1$ or $V_2$ I try \begin{array}{cc} \lambda & 0 \\ 0 & 8-\lambda \\ ...
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1answer
33 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
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1answer
66 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
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1answer
27 views

Surface Integral of the Partial Derivative of a Harmonic Function

Assume that $V$ is a solid in $\mathbb{R}^3$ which is bounded by a surface $S$ whose normal is $\overrightarrow{n}$ and $f:V \rightarrow \mathbb{R}^3$ is a harmonic function on $V$. Show that ...
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1answer
21 views

Existence of a Non-Linear Function Satisfying Certain Conditions

Let $f: \mathbb{R} \rightarrow \mathbb{R^m}$ and $N: \mathbb{R} \rightarrow \mathbb{R^m}$ be two mappings satisfying: $$\lim_{h\to0} \frac{f(a+h)-f(a)-N(h)}{h}=\vec 0.$$ If $f'(a)$ exists and is ...
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1answer
50 views

Infinity as a boundary condition - Laplace's equation

I am familiar with the uniqueness theorem that claims that given conditions on the boundary of a closed surface, if there exists a function $\varphi$ such that $\nabla^2\varphi = 0$ which also satisfy ...
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0answers
15 views

What would the phrase “attain an upper bound of the line integral” mean for vector fields?

I am working on an exercise that is asking to find the upper bound of a line integral over the unit disk where the vector field has magnitude one. I am then asked to find a vector field that attains ...
2
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1answer
53 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
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2answers
44 views

linear transformation matrix under the line integral

Is there a general methodology/approach for evaluating an integral of this form? $$ \int_C {\bf Ax} \cdot \mathrm{d}{\bf x} $$ Assume ${\bf x} \in \mathbb{R}^{n \times 1}$ and ${\bf A} \in ...
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0answers
51 views

Find a vector field G such that F = curl(G), Given F = …

Given $F = (xe^y, -x \cos z, -ze^y)$, I find that $\operatorname{div}(F) = 0$ thus there should exist a vector field $G$ such that $\operatorname{curl}(G) = F$. I run into issues finding the solution. ...
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0answers
17 views

Stokes' theorem and line integrals

Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, ...
2
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1answer
78 views

Is the Laplacian a vector or a scalar?

Need to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is ...
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1answer
28 views

Area of surface [duplicate]

What is the area of the region that is bounded by the curve $$\vec{R}(t)=(\cos^3t, \sin^3t), 0\leq t<2\pi?$$ I have no idea how to start here or what i have to use.
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1answer
37 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
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0answers
33 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
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1answer
24 views

Divergence of $\phi$ from p

I am reading a paper which is based mostly on divergence. I tried to get a basic understanding of divergence but I cannot see how it is linked with this aspect. It says: $D(\phi,p) = \phi . ...
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4answers
41 views

Difference between $\nabla T$ and $\nabla \cdot E$

Why is $\nabla T = (\frac{\delta T}{\delta x},\frac{\delta T}{\delta y},\frac{\delta T}{\delta z})$, but $\nabla \cdot E \neq (\frac{\delta E}{\delta x},\frac{\delta E}{\delta y},\frac{\delta ...
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3answers
48 views

Why is $a\times b = 0$ when $b = 2*a$?

In vector calculus, why is $a\times b = 0$ when you know that $b=2*a$? So how how do you know that a crossproduct of a vector and two times that vector is always zero?
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1answer
24 views

Integral of 2-D Laplacian

I am so confused on these integrals. Here is the question. Problem $$G(x,y)=\ln(x^2+y^2)/2$$ Calculate the 2-D Laplacian $\Delta^2G$ For the interior $D$ of the circle $C$ of radius $a$ calculate ...
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0answers
17 views

Calculating the Flux of a Surface

I am having some trouble with this problem, I feel like I am just confusing myself and I could really use some direction. Problem "For positive $a$ and $h$ let $A$ designate the region of $R^3$ ...
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1answer
16 views

Under what condition does $A^T(B \times C) + (B\times C)^T A = 2A^T(B \times C)$, A,B,C vectors

In my classical mechanics text book there is a formula that states $(\dot r_c + \omega_i \times d_i)^T (\dot r_c + \omega_i \times d_i)$ give rise to $\dot r_c^T \dot r_c + 2\dot r_c^T(\omega_i ...
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0answers
25 views

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$ I changed $x^2+y^2=4$ to $4sin^2\theta + 4cos^2\theta = 4$ so $x=2cos\theta$ and $y=2sin\theta$ and then ...
1
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1answer
35 views

Closed surface integrals

Can somebody give me hints to solve the following question? I need to find the closed surface integral (using divergence theorem) of $$\oint \vec{r} (\vec{a} \cdot \vec{n}) da$$ where $\vec{n}$ is ...
2
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2answers
27 views

What will be value of $\vec{r} \cdot \nabla$

I was studying on Nabla Operator and saw that $\nabla \cdot \vec{r} \neq \vec{r} \cdot \nabla$ So, if I were to find $\vec{r} \cdot \nabla$ how would I calculate it? I know that $\vec{r} \cdot ...
2
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2answers
37 views

Divergence and Curl (involving constant vectors)

How find the divergence and Curl of the following: $(\vec{a} \cdot \vec{r}) \vec{b}$, where $\vec{a}$ and $\vec{b}$ are the constant vectors and $\vec{r}$ is the radius vector. I have tried solving ...
3
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1answer
29 views

Divergence and Curl of the vectors

How to find the divergence and the curl of the given vectors? a. $( \vec{u} \cdot \vec{r}) \vec{v}$ b. $( \vec{u} \cdot \vec{r}) \vec{r}$ c. $( \vec{u} \times \vec{r})$ d. $ \vec{r} \times(\vec{u} ...
2
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0answers
24 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
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0answers
18 views

Derivation of centrifugal acceleration with Coriolis theorem

Is there a way to derive the centrifugal acceleration of an object rotating with a constant speed $V$ on a circle with radios $r$ with Coriolis theorem ?? thanks
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0answers
40 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
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1answer
18 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
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1answer
50 views

If $\alpha$ is a unit speed curve of constant curvature lying in a sphere, then $\alpha$ is a circle.

I'm trying to solve the following problem but got stuck along the way. I would like some help on getting this through. Prove that if $\alpha$ is a unit speed curve of constant curvature lying in a ...
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2answers
31 views

If $\vec p\times((\vec x-\vec q)\times\vec p)+\vec q\times((\vec x-\vec r)\times\vec q)+\vec r\times((\vec x-\vec p)\times\vec r)=0$, find $\vec x$

Let $\vec p, \vec q$ and $\vec r$ are three mutually perpendicular vectors of the same magnitude. If a vector $\vec x$ satisfies the equation $\begin{aligned} \vec p \times ((\vec x - \vec q) \times ...
3
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3answers
100 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
1
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3answers
50 views

Finding a point a certain distance away from 2 points

I need to find a point that is a certain distance away from two known points. Where $P_1, P_2, L_2$ and $L_1$ are all defined and that is all that is known. How do I find $P_3?$ Kind Regards.
2
votes
1answer
46 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
1
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2answers
41 views

Question in vector algebra regarding minimum value of modulus.

If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three coplanar unit vectors such that $\vec{a} +\vec{b} +\vec{c} =0$. If three vectors $\vec{p}$ , $\vec{q}$ , $\vec{r}$ are parallel to $\vec{a}$ , ...