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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18 views

Line integral with differentials (cylindrical/spherical)

How can I write a line integral of a vector field with exact differentials in cylindrical and spherical coordinates? I know for in cartesian coordinates: $$ \mathbf{E}(x,y,z)=P\mathbf{\hat{x}}+Q\...
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0answers
9 views

Super shape (formula) normal vector gives wrong answer for the case of a ellipse and circle?

Im trying to derive the normal vector a super shape, I took this approach: $$ r(\theta) = (|\frac{1}{a}cos(\frac{m\theta}{4})|^{n_2}+|\frac{1}{b}sin(\frac{m\theta}{4})|^{n_3})^{\frac{-1}{n_1}}$$ I ...
4
votes
1answer
57 views

Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
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1answer
20 views

Gradient of a maximum

How do you compute the gradient of a function that involves a maximum? For example, I have the function: $$ f(\vec{t}) = v(1-\exp(-\lambda\cdot \max(t_0,t_1)))$$ With $v$ and $\lambda$ constant, for ...
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0answers
9 views

Divergence Theorem: Conditions for the boundary integration to vanish?

Consider the Divergence Theorem for example in two dimensions, in the upper right quadrant of Euclidean space: $$\int_0^\infty dx \int_0^\infty dy ~\vec\nabla\cdot\vec F=\oint_C ds~\vec n\cdot\vec F$$...
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1answer
27 views

Integral of solid angle of closed surface from the exterior

Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, ...
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0answers
13 views

Fourier transform of “nabla matrix”

I'm reading the paper: http://projecteuclid.org/euclid.cmp/1103941230. I cannnot understand a sentence after the equation (9), that is: Therefore the Fourier transform of $\nabla u$ and $\omega$ ...
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2answers
27 views

Space curves in Vector Analysis and proofs

given the space curve $x=a\cos t$, $y=a\sin t$, $z=bt$ show that $k=\dfrac{a}{(a^2+b^2)}$ My solution : Assuming $a,b$ and $c$ are positive: $k(a^2+b^2)=a$ $0\leq t\le2\pi$ $X^2+y^2=a^2$ ...
2
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2answers
54 views

Why isn't the gradient vector of a parametric curve parallel to the tangent vector?

Consider a parametric curve defined by the equation: $$\mathbf{r}(t) = X(t)\mathbf{\hat{i}} + Y(t)\mathbf{\hat{j}} + Z(t)\mathbf{\hat{k}}$$ Paul's online math notes indicate that the unit tangent ...
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0answers
18 views

How to shorten dot product

I would like to shorten a dot/scalar product: $$f(s)=sP_1+s^2P_2+\big((P_2-P_1)^TsN_1\big)N_1$$ Here $s$ are scalars, $P$ are points and $N$ are unit normal vectors in $R^3$. The function $f(s)$ ...
3
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1answer
56 views

Working with Lagrange multipliers, reducing gradients is okay, right?

I am employing the method of Lagrange multipliers to determine a maximum. As part of this, I arrive at the following equation involving two gradients and the parameter $\lambda$, as is common for ...
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2answers
39 views

If $\vec{u} = \vec{r}/r$, find $\mathrm{div} ~(\nabla \vec{u})$?

If $\vec{u} = \vec{r}/r$, find $\mathrm{div} ~(\nabla \vec{u})$. Please help me to solve this problem.
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1answer
58 views

How to prove this version of the fundamental theorem of calculus for curves in the closure of a domain

Dear Downvoters: if you leave a comment, you can influence the way this post gets modified, if you don't this post might never satisfy you - even though I keep editing Let $\Omega \subseteq \mathbb{...
1
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2answers
41 views

Different notation for position vectors? Domain/Range?

What is the difference between this notation for position vectors? Are there any differences in domain and range? $$ \mathbf{r}=x{\mathbf{\hat{e}}_x}+y{\mathbf{\hat{e}}_y}+z{\mathbf{\hat{e}}_z} \qquad ...
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0answers
13 views

Curl pde with Dirichlet boundary condition in a simply connected domain

Let $\Omega$ be an open, bounded, connected, simply connected domain in $\mathbb{R}^3$, with boundary $\partial\Omega=\partial\Omega_1\cup \partial\Omega_2$. Suppose $\mathbf{H}\in H(\mbox{curl};\...
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1answer
24 views

Converting from partial derivatives of $f(r)$ in $x, y, z$ to dot-product of vectors

This is from my mathematical physics book. I don't know how the right side arrived from the left side of the equation: $$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} + z\frac{\...
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1answer
22 views

Find the vector v that has norm equal to 3 and has the same direction as the vector <0,1,-1>

What I did was normalized v, which gives $v=\sqrt{0^{2}+1^{2}+-1^{2}}$ then I divided that by the norm of the vector with the same direction so that $u=\sqrt{2}/3$ and multiply that by vector v's ...
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1answer
73 views

Solving $\oint_c \textbf{F}\centerdot d\textbf{r}$ using Stokes' Theorem

It would be great if someone can help me with this problem: $S$ is the graph of \begin{equation*} f(x,y)=4x-8y+30 \end{equation*} Over the rectangle $R$ \begin{equation*}R=\left \{ (x,y)|-2<x<3,...
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1answer
24 views

Surface integral with vector integrand identity

On page 155 of G.E. Hay Vector and Tensor Analysis, he asks the reader to prove that $$\int_S \mathbf{n}\times\mathbf{x}\ dS = 0$$ where $\mathbf{n}$ is the unit outer normal and $\mathbf{x}$ is the ...
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1answer
62 views

help with vector calculus [closed]

the question is : how do I prove that: $\nabla^2 (r^n\vec r)=n(n+3)r^{n-2}\vec r$
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2answers
50 views

evaluating curl of $\vec r/r^2$

how do I calculate curl of : $\vec r/r^2$ I don't know how to solve this problem can someone help me please
2
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0answers
34 views

Zorich's Mathematical Analysis, Volume II

Springer just published a new English version of Vladmir Zorich's two-volume Mathematical Analysis. I was looking at the second volume. It seems to have sections on both Multivariable/Vector Calculus ...
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0answers
22 views

Uniqueness theorem regarding Helmholtz decomposition of a Vector field

Helmholtz theorem wiki link states that given a smooth vector field $\pmb{H(x,y,z)}$, there are a scalar field $\phi (x,y,z)$ and a vector field $\pmb{G(x,y,z)}$ such that $$\pmb{H}=\pmb{\nabla} \phi +...
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1answer
18 views

Two Dimensional Motion using Vector Analysis

I came across the following question: Use vector methods to find the maximum angle to the horizontal at which a stone may be thrown so as to ensure that it is always moving away from the thrower....
2
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2answers
20 views

Evaluate the flux of the vector field $\vec F = -9\hat j- 3 \hat k$ on the surface $z=y$ bounded by the sphere $x^2+y^2+z^2=16$

Evaluate the flux of the vector field $\vec F = -9\hat j- 3 \hat k$ on the surface $z=y$ bounded by the sphere $x^2+y^2+z^2=16$ My attempt: $$\iint_S \vec F \cdot \vec n dS = \iint_S (0,-9,-3) \...
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0answers
23 views

Position vector, vectorfunction or vector field?

This notation confused me. The position vector in the wikipedia-article is denoted (cartesian coordinates): $$\mathbf{r}(t)=\mathbf{r}(x,y,z)=x(t)\mathbf{\hat{e}}_x+y(t)\mathbf{\hat{e}}_y+z(t)\mathbf{...
0
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1answer
39 views

Existence and Smoothness of Vector Calculus Identities [closed]

How to proof there exist smooth and globally defined solutions to all Vector calculus indentities ? For example: proof there exist smooth and globally defined solutions to the divergence of the curl ...
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1answer
52 views

Solve the vector x:

Solve the vector $\mathbf{x}$: $$\mathbf{x}\times \mathbf{\beta}=\mathbf{r},$$ $$\mathbf{x}.\mathbf{\alpha}=3,$$ where $\mathbf{\alpha}=\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{\beta}=2\mathbf{i}-\...
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1answer
30 views

Basis Vectors in a General Curvilinear Coordinate System

I'm confused as to how does one find out the basis vectors of a curvilinear co-ordinate system. In the context of a general, arbitrary curvilinear co-ordinate system, the textbook I'm reading states ...
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0answers
15 views

Luminosity and Apparent flux

The stars in our Galaxy have luminosities ranging from $L_{\text{min}}$ to $L_{\text{max}}$. Suppose that the number of stars per unit volume with luminosities in the range of $L$, $L+dL$ is $n(L)dL$. ...
0
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1answer
35 views

Divergence of a vector field in an orthogonal curvilinear coordinate system

How would one go about proving the following result in $\mathbb R^3$ for the divergence of vector field $\vec F = F_i \hat e^i$ $$ \nabla \cdot {\mathbf F} = \frac{1}{h_1 h_2 h_3} \left[\frac \...
1
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1answer
27 views

Killing/Isometry correspondence: Domains of flows generated by vector fields

I am wondering about the correspondence between the isometry group $\mathcal{I}$ and the Lie Algebra of Killing vector fields $\mathcal{K}$ on a pseudo-Riemannian manifold $(\mathcal{M}, \mathbf{g})$. ...
1
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1answer
38 views

Equivalent form of a vector area of a surface

I am interested in showing that the vector area $$\int_{\mathcal{S}}da$$ can be equivalently given by $$\int_{\mathcal{S}} da = \frac{1}{2}\oint(r \times dI).$$ I am mostly interested in getting a ...
2
votes
2answers
54 views

Proof of $\vec{\nabla}\cdot(\vec{\nabla}^2\vec{F}) = \vec{\nabla}^2(\vec{\nabla}\cdot\vec{F})$

How can I prove the following? $$\vec{\nabla}\cdot(\vec{\nabla}^2\vec{F}) = \vec{\nabla}^2(\vec{\nabla}\cdot\vec{F})$$ $$\vec{F}:\Bbb{R^3}\mapsto\Bbb{R^3}$$ I am confused because on the left part I ...
1
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0answers
35 views

Gradient of a function involving maxima

How do I find the gradient of a function like $f(\vec{v})$ where $$ f(\vec{v}) = \max_{\vec{t}\geq 0} g(\vec{v},\vec{t})$$ For example, I have a function defined as follows: \begin{align} f(\vec{v}) ...
0
votes
1answer
20 views

Solution to specific surface integral involving a projection

I have an integral computation question: Given the vector field $v = y \hat{z}$ I want to compute the surface integral $\int (\nabla \times v) \cdot da$ of the surface within the triangle with ...
0
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1answer
24 views

Steps used to calculate the partial derivative x-component used to calculate divergence?

Given a spherically symmetric vector field with amplitude increasing as the square of the distance from the origin. Thus $$\vec A=r^2\hat r$$ $$r^2=(x^2 + y^2 + z^2)$$ $$\hat r =\frac {x\hat i+y\...
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1answer
58 views

Einstein notation difficulties

I'm just learning the Einstein index notation, and came across this derivation in a textbook. I couldn't follow the steps. Can someone please help me out? The first order differential equation: $$\...
1
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0answers
25 views

Total gradient of height field f(x,y,z)=0

I have a function given by $$f(x,y,z) = 0$$ I sample this function in a tangent plane around a given point $A = (x,y,z)$. I create N points $A_i$ within a disc neighborhood of a point $A$ in its ...
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0answers
21 views

A discrepancy in a vector identity involving gradient of product of 2 vectors.

We know that: $\nabla (Α\cdotΒ)=Α\times (\nabla \times Β)+Β\times (\nabla \times Α)+(Α\cdot\nabla )Β+(Β\cdot\nabla )Α$ Where $A$ and $B$ are two vectors. Now, suppose that the curl and divergence ...
3
votes
3answers
69 views

what does $(A\cdot\nabla)B$ mean?

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this? I've also seen this in some identities
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0answers
22 views

Gradient of 3d delta-function

I need to evaluate the following expression: $\int \mathrm{d}\boldsymbol{r} \left[\nabla_{\boldsymbol{R}_\alpha}\delta(\boldsymbol{r}-\boldsymbol{R}_\alpha)\right]v(\boldsymbol{r})$ and I want to ...
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0answers
11 views

Nonlinear Conjugate gradient for vector valued multi variable functions?

So far what I have found in online and in Numerical recipes book describe algorithm for scalar value multi variable function. Can anyone point me to the algorithm for nonlinear conjugate gradient for ...
1
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1answer
20 views

Potential of vector field is undefined on Y-axis although field is defined

I'm having the following vector field: $$\vec{F}(x,y) = (\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$$ The field is conservative in $\mathbb{R}^2 \backslash (0,0)$ as long as your curve doesn't encircle $(...
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0answers
26 views

Confusion with chain rule when proving statement about tangent plane to a point in a manifold

I'm trying to prove the following: If $f:\mathbb{R}^3 \to \mathbb{R}$ is a differentiable function, $a \in \mathbb{R}$ is a regular value of $f$ and $S=f^{-1}(a)$, then for all $p \in S$ the tangent ...
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1answer
33 views

curl and stokes application

I cannot fin the flux of $$F(x,y,z)=(y^2cos(xz),x^3e^{yz},-e^{-xyz})$$ through the portion of sphere $$\Sigma = \{x^2+y^2+(z-2)^2=8, z\ge0 \}$$ I think Stokes th. must be used, so in spherical ...
1
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1answer
45 views

Calc 3: Calculate Work Done on Particle [closed]

I've been working on this problem for a while and I'm pretty stuck. I tried it multiple different ways, by the last time I attempted it I realized that I hadn't converted kilometers to meters the ...
0
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2answers
39 views

Solving vector equation 3

Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get $1+\bar{a}.\bar{y}=|...
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1answer
35 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
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1answer
57 views

Solving vector equation 1 [closed]

Using vector method solve $p \bar{x}+\bar{x}(\bar{x}.\bar{b})=\bar{a}\times \bar{b}+\bar{c}$ How to solve $\bar{x}$ from such vector equation. Please help.