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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Gaussian theorem: determining a surface and exterior normal from solid figure

Given a vector field: $ \vec{v} = (z^2-x^2-y^2-2) \cdot (x,y,z) $ and a the solid figure $ K = \{(x,y,z) \in \mathbb{R} : z > \sqrt{2+x^2+y^2}, 2 < z < 3 \} $. To determine: $ \int_{K} ...
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1answer
18 views

Why is the expression $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$?

It says in my lecture notes that the index notation $u_{j}\frac{\partial u_{j}}{\partial x_{i}}$ is equivalent to $\frac{1}{2}\frac{\partial}{\partial x_{i}}(u_{j}u_{j})$, but does not explain why. ...
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1answer
35 views

To evaluate using Gauss Divergence Theorem

Using Gauss Divergence Theorem, evaluate the integral $\int_{S}\int F.\hat n dS$ where $F=(4xz,-y^2,4yz)$ . S is surface of solid bounded by sphere $x^2+y^2+z^2=10$ and paraboloid $x^2+y^2=z-2$ and ...
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0answers
18 views

$F=(y^2z^2,z^2x^2,x^2y^2)$ be a vector field , to find a nonzero scalar field $f$ such that $fF$ is a grdient i.e. $curl(fF)=O$?

Let $F=(y^2z^2,z^2x^2,x^2y^2)$ be a vector field . how to find a nonzero scalar field $f$ such that $fF$ is a grdient i.e. $curl(fF)=O$ ? Please help . In general , given a vector field $F$ in ...
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1answer
19 views

Application of Gauss Divergence Theorem

Consider $$-\triangle u = f \ \ \ \ \ \text{in} \ \ \ \Omega$$ .$$\frac{\partial u}{\partial n} = g \ \ \ \text{on} \ \ \ \ \partial \Omega $$ Where $\Omega \subset ...
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2answers
130 views

What is the normal vector to the plane ax+by+cz=d?

If d=0, then we obviously see that the equation is the Euclidean inner product with (a,b,c) and (x,y,z) that equals zero - and so (a,b,c) is the normal vector to the plane. What if $d \ne 0$? Then ...
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2answers
39 views

Strong maxima and minima

I'm stuck with this problem, in particular at b): Let $u:D \subseteq \mathbb{R}^2 \to \mathbb{R} \in C^2$ a harmonic function. $u$ has a local maximum at point $\vec{p} \in D$. Then: (a) Show that, ...
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1answer
22 views

Equivalence of two vector fields

Let $V$ be a convex region in $\mathbb R^3$ whose boundary is a closed surface $S$ and let $\vec n $ be the unit outer normal to $S$. Let $F$ and $G$ be two continuously differentiable vector fields ...
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47 views

Gauss´s law proof “details”

I know that this question has already been asked multiple times but I´m still not getting on the mathematical details behind the answers... So I hope that this question doesn´t get closed; also I ...
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1answer
28 views

Regarding gauss law differential form

I have a big issue regarding the equality of integrands in gauss law. Given the integral form we have that $$\oint_{\partial\Omega}\vec{E}\cdot\vec{dS}=\int_{\Omega}\nabla\cdot \vec{E}dV={1\over ...
2
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0answers
32 views

$\nabla^2 u = 0 $ and integral $u$ around $\partial B_\rho$

I'm doing exercises from book about vector calculus. And there is problem wich I'm not sure what to do. Let's see the hypotesis. Let, $D \subseteq \mathbb{R}^2$ an open set. Let, $u:\overline{D} \to ...
2
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2answers
39 views

How to differentiate the numerator of this vector field $\frac{\vec r}{|r|^3}$?

I was studying a nice solution of how to apply the divergence theorem for a vector field with a singularity at the origin. However, the solution doesn't give any concrete computations and just makes ...
2
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2answers
38 views

Does there exist a vector field $\vec F$ such that $curl \vec F=x \vec i+y\vec j+z \vec k$? [closed]

Does there exist a vector field $\vec F$ such that curl of $\vec F$ is $x \vec i+y\vec j+z \vec k$ ? UPDATE : I did $div(curl \vec F)=0$ as the answers did ; but that assumes a lot i.e. it assumes ...
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1answer
41 views

Derivative of $f(X)=X^T$

Suppose that $X$ is a vector in $R^n$ and $f$ is a function that receives $X$ and gives $X^T$ which is the transpose. Is $f$ differentiable? If yes, what is its derivative?
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2answers
35 views

Vector analysis: understanding formulas for normal and tangent

I'm studying a chapter on vector analysis and I don't really understand why the following equations are switched for the two situations. (My course is not in English, so I apologize if I translated ...
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3answers
35 views

How to understand the equality about $(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} $?

For the relation, $$(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} = (\mathrm{curl}\mathbf{q})\times\mathbf{q}+\frac{1}{2}\mathrm{grad}|\mathbf{q}|^2,$$ is there any physics, geometry, or basic intuitive ...
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1answer
42 views

curved space vs linear space with curved basis

What is actually the difference between a curved space and an euclidean space represented in curvilinear basis?
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1answer
39 views

Sum of two vectors in spherical coordinates. [duplicate]

What is the sum of two vectors in spherical coordinates? The coordinate system: Assume we have vectors $(r_1,\theta_1,\phi_1)$ and $(r_2,\theta_2,\phi_2)$ in spherical coordinates. I know the sum ...
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0answers
69 views

Differences between solenoidal and rotational vector fields

In classification of vector fields, one of the 4 different type vector fields is "solenoidal and irrotational vector field" (both divergence-free and curl-free). If solenoidal and rotational vector ...
3
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1answer
58 views

Why is divergence defined as $\mathbf{\nabla} \cdot \mathbf{v}$?

Suppose I am working in $\Bbb R^3$. Suppose I have a pond and I drop some dust on the surface. If the materials spread out, I have positive divergence, usually. Let $\mathbf{v}(\mathbf{x})$ denote ...
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2answers
27 views

Prove that divergense and curl free vector field is a constant vector field

I need to prove the fact that a vector field $\vec{B}$ that is divergence and curl free, is a constant vector field. I have attempted to prove this by referring to the divergence, but realized that ...
2
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1answer
29 views

Einstein Summation with Del Operator

Can someone show explicitly me why $2B_k\nabla B_k = \nabla B^2$ ? Is $B_k\nabla B_k$ just $B_x\nabla B_x+B_y\nabla B_y+B_z\nabla B_z$? But then I end up with nine terms on the LHS and I can't ...
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1answer
36 views

Vector space of vector fields [closed]

Is the set of all vector fields (on $\Bbb R^3$ for instance) a vector space? What about the set of all continuous vector fields on $\Bbb R^3$? It seems that this is just a generalization of the ...
3
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0answers
17 views

Gradient control by div and curl

For any $u\in W^{1,p}(\mathbb R^3)$ how to prove: $$\|\nabla u\|_p \leq C(\|\operatorname{div}u\|_p+\|\operatorname{curl}u\|_p)$$ Any suggestions? Thanks!
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2answers
47 views

Does the gradient of the gradient make sense?

We know that for a vector-valued function $A$ we can define $$\operatorname{curl}(\operatorname{curl}(A))$$ where the curl operator is $\operatorname{curl}(A) = \nabla \times A$. But given an ...
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1answer
47 views

Prove if $\epsilon>0$ is sufficiently small then $f(\underline{p}+\epsilon\underline{n}\left(\underline{p}\right))>0$

A region in $\mathbb{R}^n$ is a subset $\Omega$ of $\mathbb{R}^n$ for which $\exists$ a function $f:\mathbb{R}^n\ \rightarrow\ \mathbb{R}$ with following properties: (i) all partial derivatives of f ...
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0answers
62 views

What is the advantage of using Feynman's trick to use rules of vector algebras on $\bf \nabla$ operator?

I was reading Field Energy & Field Momentum of Feynman's Lectures on Physics Vol.II; there he introduced a trick 'to throw out—for a while at least—the rule of the calculus notation about what the ...
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3answers
48 views

Identifying the basis a vector is written in

Given some vector in a general vector space and the coefficients corresponding to this vector written as a linear combination of some orthonormal basis vectors, is it possible to determine the basis ...
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34 views

I don't no undertand of Doppler effect.

When a sound with frequency f is produced by an object traveling along a straight line with speed u and a listener is traveling along the same line in the opposite direction with speed v, the listener ...
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1answer
50 views

Boundary on $R^3$ about Stoke's theorem.

Let S be the union of two surfaces, $S_1$ and $S_2$, where $S_1$ is the set of $(x,y,z)$ with $x^2+y^2 =1$ ,$0 \le z \le 1$ and $S_2$ is the set of $(x,y,z)$ with $x^2+y^2+(z-1)^2 =1$, $z \ge 1$. Set ...
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2answers
42 views

Using Divergence Theorem to evaluate the flux over a sphere

Above is the question. I've try to find the divergence of F and parameterize the sphere using spherical coordinates. Below is my work. Then I use online integral calculator(just to avoid human ...
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3answers
36 views

To prove that a vector $x(t)$ lies in a plane.

Prove that vector $x(t)=t\,\hat{i}+\left(\dfrac{1+t}{t}\right)\hat{ j}+\left(\dfrac{1-t^2}{t})\right)\hat{k}$ lies in a curve. I am puzzled. Don't know how to approach it.
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30 views

Gradient of distance between dot product of the distance between two vectors

I'm wondering how to compute the following gradient. $\bigtriangledown ((r-r_1)\cdot (r-r_1))$ where r and $r_1$ are vectors. Is there always a way I could compute this gradient using index ...
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0answers
59 views

Gradient in geometric calculus

In geometric calculus I see that we can unify the three fundamental derivatives from vector calculus; the gradient, the curl, and the diverge; into one operator. However, $\nabla$ is defined for any ...
4
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1answer
47 views

What is the geometrical meaning of the total differential?

Could anybody give me a geometrical explanation of the total differential, if there is such? For me(a non-mathematician) it just looks like the generalization of the derivative to more dimensions but ...
3
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2answers
63 views

Is $(\nabla \times \nabla)$ an operator?

Is $(\nabla \times \nabla)$ an operator? I am wondering if its possible to compute \begin{align} \vec{f} \cdot (\nabla \times \nabla) \end{align} Where $f$ is a vector. I would be interested in both ...
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0answers
17 views

Gradient of euclidean norm of a vector field

Does $\nabla \| \boldsymbol{v} (\boldsymbol{x}) \| = \frac{1}{\| \boldsymbol{v}(\boldsymbol{x}) \|} \frac{1}{2} \left( \nabla \boldsymbol{v}(\boldsymbol{x})+\nabla \boldsymbol{v}^T (\boldsymbol{x}) ...
4
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1answer
62 views

Divergence operator of higher order and intrinsic point of view

Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that : $\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, ...
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0answers
11 views

find the flux by using spherical polar coordinates

Q: Using spherical polar coordinates, find the flux of the vector field F = (y, x, z) out of a sphere of radius a centred at the origin. My thinking: the question is asking for ∫∫Fn dS. using ...
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0answers
22 views

formula for getting the cross product in spherical coordinates, given the two vectors.

I am using this coordinate system: consider these two vectors in spherical coordinates: $$\vec A=A_r \hat r +A_\phi \hat \phi + A_\theta \hat \theta= A_r\sin{\theta_A} \cos{\phi_A} \hat i ...
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1answer
19 views

Calculate the work of the force generated by an electric charge in movement

We know that the force generated by an electric charge that is located at the origin, on a charged particle at a point $(x,y,z)$ of position vector ...
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1answer
44 views

What does the gradient of the gradient ($\nabla\nabla u$) mean?

For a scalar function $u$ defined on $\mathbb{R}^n$ we have this equality: $$\nabla(|\nabla u|^{p-2})=(p-2)|\nabla u|^{p-4}\nabla u\nabla\nabla u$$ My question is : what does $\nabla\nabla u$ mean? I ...
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2answers
35 views

How do I evaluate the following vector product

I need to evaluate the following $$((\vec{a}\times\vec{b})\times\vec{a})_i((\vec{a}\times\vec{b})\times\vec{a})_j.$$ I am assuming Levi civita notation would be useful, but couldn't utilise it. Does ...
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1answer
90 views

Cauchy Integral Formula Questions

(i) $$\int_{|z|=4}^{} \frac{e^{3z}}{z-i\pi} dz$$ Using Cauchy's integral formula I get $f(z)=e^{3z}, 2\pi if(i \pi)=-2\pi i$ Is this correct? (ii) $$\int_{|z|=4}^{} \frac{e^{3z}}{(z+i\pi)^7} dz$$ For ...
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0answers
9 views

How does the Fréchet derivative link to the gradient $\Delta$?

Forgive me if I'm being stupid, but from my analysis textbook it seems like the Fréchet derivative and the gradient of a vector-valued function are the same thing, however they are presented as being ...
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2answers
19 views

Finding a potential

$F(x,y)= \left(\displaystyle\frac{1-y^2}{(1+xy)^2},\displaystyle\frac{1-x^2}{(1+xy)^2}\right)$. I've been having some troubles to find the potential of $F$. To find it the idea was finding $\int ...
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1answer
40 views

Where is f complex differentiable

Let $f(z):=2|z|^2-\overline{z}^2$ where $z\in \mathbb{C}$ At what points is f complex differentiable? I can assume $\lim_{h\to0} \frac{\overline{h}}{h}$ does not exist. I have $$\lim_{h\to0} ...
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1answer
26 views

cross product of E and H

Can anyone explain how they came up with the product of E and H ? I don't understand why the exponent of E cross H are multiplied by 2. Thanks
1
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1answer
27 views

Rate of change and gradient

I'm trying to solve the follow problem: Suppose that we are on the point $P=(1/\sqrt{2},1/2,1/2)$ over $z=\sqrt{1-x^2-y^2},$ $z\geq 0, x^2+y^2<1.$ In which direction we have to move over the ...
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2answers
117 views

The “inverse” of $\nabla\times$ operator

From physics, just to use a well known example, we know that the relationship between the magnetic induction $\mathbf{B}$ and the potential vector $\mathbf{A}$ is given by: $$\mathbf{B} = ...