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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2answers
73 views

Gauss Divergence theorem gives a wrong result for a surface integral

Evaluate $\iint _{{S}}(y^{2}z^{2}i+z^{2}x^{2}j+x^{2}y^{2}k)\cdot \,ds$ where $S$ is the part of the sphere $x^{2}+y^{2}+z^{2}=1$\, above the xy-plane. Answer to this question is $\pi/24$ as ...
1
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1answer
93 views

Prove $\nabla u \times \nabla v =0$ is a necessary and sufficient condition that $u$ and $v$ are functionally related by the equation $F(u,v) = 0$

Let $u$ and $v$ be differentiable functions of $x$, $y$ and $z$. Show that a necessary and sufficient condition that $u$ and $v$ are functionally related by the equation $F(u,v) = 0$ is that ...
5
votes
2answers
139 views

How does curl relate to rotation?

The operation mathematically means $$(\nabla \times \vec A)\cdot\hat n = \lim_{\Delta S\to\ 0} \frac{\oint\vec A\cdot\ d\vec l }{\left | \Delta S \right |}$$ and the proof of this is quite logical. ...
1
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1answer
67 views

Vector calculus - Material derivative in spherical coordinates…

So this one might be a little simple for some of you but I was hoping I could get all the nuts and bolts needed to show this for myself. I have the following relationship, which makes use of the the ...
1
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1answer
22 views

Connect two points in $\mathbb R^3$ using integral lines of some vector fields

Suppose we are in $\mathbb R^3$ and let $\nu\in S^2$ be fixed. We have a regular function $u:\mathbb R^3\to \mathbb R$ with the following property: $u$ is decreasing along the integral lines of the ...
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0answers
24 views

At what angle doe the line $2x = y = 2z$ intersect the ellipsoid $2x^2 + y^2 + 2z^2 = 8$?

What does the angle between a line and an ellipsoid mean? Does it mean the angle between the line and a plane tangent to the ellipsoid at the point of intersection? By making this assumption, I end ...
0
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0answers
90 views

How to compute a surface integral of a vector field over a closed surface, if the field is divergence-free?

I've have come across a problem like this a couple times now and don't really understand how to arrive at a non-trivial answer. By equating the surface integral of the vector field over the closed ...
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1answer
30 views

$\{(x,y) \in \mathbb{R^2} : x^2+y^2 > c\}$ is neither open nor closed? [closed]

Why $$\{(x,y) \in \mathbb{R^2} : x^2+y^2 > c\} \space (c > 0) $$is neither open nor closed?
1
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1answer
63 views

How to interpret the integrand in this surface integral?

Let Ω be the region in $ℝ^3$ defined by $$ Ω={(x_1,x_2,x_3):max(∣∣x_1∣∣,∣∣x_2∣∣,∣∣x_3∣∣)≤1}$$ Let ∂Ω denote the boundary of Ω. Calculate $$∫_{∂Ω}ϕF⋅ndσ$$ where n is the unit normal vector, dσ ...
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2answers
56 views

Extreme Values of $f(P)$ When $P$ Lies Inside a Triangle with Vectices $A$,$B$, $C$

I would appreciate if somebody could help me with the following problem Q:$P(x,y)$ given point lie inside or on the boundary of triangle $\triangle ABC$. Find maximum and minimum $f(a,b,c)$ ...
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1answer
26 views

Vector relationships

I'm a beginner with vector calculus and analysis and I am looking at the following example problem. Please note that is is not a homework question, I am simply looking at some vector relationships. ...
1
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1answer
24 views

Finding a scalar potential from

So I have a force given by: $$ F = (x^2 + y^2 + z^2)^n(xi+yj+zk)$$ From this I want to find a scalar potential (defined as $\phi(x,y,z)$) so that $F = -\nabla\phi$. Can anyone give me some pointers ...
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2answers
39 views

Curl of a vector field: [closed]

So I have a force given by $$ F = (x^2 + y^2 + z^2)^n(xi+yj+zk)$$ I was wondering how we handle this for the curl...an explanation of the setup would be excellent.
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2answers
21 views

Divergence Example in 3D Cartesian Coordinates

I've just started my Galaxy Dynamics module and we are refreshing ourselves on Divergence and Curl, etc etc. I've come across a divergence example which I can't quite understand how to handle: So I ...
2
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2answers
71 views

Interpreting partial derivatives $\frac{\partial f}{\partial u}$ in differential geometry

I'm reading Do Carmo's "Riemannian Geometry" and at some point he introduces the following notation: Let $A \subset \mathbb{R}^2$ be an open region bounded by a piecewise differentiable curve and ...
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2answers
34 views

Difficult integral evaluation

I'm working through Vector Calculus by Marsden and Tromba to review for my GRE (and because it has really interesting historical snippets) and I ran into a wall on a problem where I have to evaluate ...
2
votes
2answers
49 views

cylindrical coordinates - base vector integral

Hello I have a problem where $\hat{p}$ is a basevector in a cylindrical system $$\int_{0}^{\pi/4}\hat{p} d\theta$$ I know that; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ ...
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1answer
12 views

How would you derive these properties using component form?

I'm currently looking over lecture notes and am looking at the following properties: Can anyone explain how they might be derived using component form?
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0answers
34 views

How to interpret the gradient of a vector field, rather than a scalar field,

If $\vec u(y,t)$ is a mapping from $R^{n+1}$ to $R^n$, then I can write out the n component functions of U, each as a function from $R^{n+1}$ to $R$ $$\vec u(y,t)=(u_1(y_1,...,y_n,t), ... ,u_n(y_1, ...
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0answers
32 views

Can I use Part 1 to solve Part 2 in this multivariable analysis problem?

Consider a function $\vec u(y,t)$:$R^{n+1}→R^n$ and a coordinate transformation $\vec x(α,t)$:$R^{n+1}→R^n$ which also depends on time t. Let u and x be related so that $\large ...
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0answers
37 views

Can this problem be solved without the assumption that f is C^2?

EDIT: I noticed that this question was downvoted twice today, so I decided to edit and add more context, e.g., the full problem statement that I am working on and have proved, although I used an ...
0
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1answer
35 views

Sequence of lowersemmi continuous functions

Given a sequence of lowersemi continuous functions $f_n : (0,1)\rightarrow [0,\infty)$ with $$\sup_n Varf_n<\infty,\ \ \ \int_0^1f_n=1.$$ Here, $Varf_n$ is the point-wise variation of $f_n$, that ...
0
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1answer
16 views

Showing properties of unit tangents

I'm working my way through the book Vector Calculus by Marden and Tromba and I've run into a bit of trouble on one of the problems (more like I'm clearly misunderstanding them but I'm not sure how). ...
0
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1answer
42 views

Curve with finite length

Let $u: [a,b]\rightarrow \mathbb{R}^d$ is a curve. We define $$Var_{[a,b]}u=\sup\{\sum_{i=1}^n|u(x_i)-u(x_{i-1})|: a\le x_0<x_1<\ldots<x_n\le b, n\in\mathbb{N}\}$$ I am trying to find a ...
0
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1answer
107 views

How to find the equation of the plane passing through the intersection of two other planes and whose perpendicular distance from the origin is given?

I've been trying to find a way to work this out for hours now with no luck. The question is: Find the equation of the plane(s) passing through the intersection of the planes $x+3y+6=0$ and ...
1
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1answer
36 views

If u is a solution of $\nabla^2 u =p(\boldsymbol{x})u$ show that u is unique [closed]

Let D be a bounded region in $\mathbb{R}^3$, and suppose $p(\boldsymbol{x})>0 $ ON $D$. If u is a solution of $\nabla^2 u =p(\boldsymbol{x})u$, $x \in D$ and $\nabla u \cdot n=g(\boldsymbol{x})$, ...
0
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0answers
9 views

If $\delta$ is a derivation at $p \in M$, then there exists a unique $v \in \mathbb{R}^n$ such that $ \delta = D_{p,v}$.

Let $M$ be a smooth manifold of dimension $n$, and let $\delta$ be a derivation at a point $p \in M$. Then, show that there exists a unique vector $v \in \mathbb{R}$ such that, $ \delta = D_{p,v}$, ...
2
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1answer
35 views

Differentiate a function with respect to a vector

I'm reading my course on quantum mechanics and I just noticed something strange (or at least new to me) that I don't really understand : $$\psi(\vec{r},t) = \psi_0 e^{\frac{i}{\hbar}(\vec{p}.\vec{r} ...
0
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1answer
17 views

Units of flow from a line integral

Given a velocity field $\vec F(x,y)$, the flow along a curve $C$ is given by $$\int_C \vec F\cdot \vec T ds= \int_C \vec F\cdot d\vec r,$$ where $\vec r(t)$ is a parametrization of $C$. What the ...
3
votes
2answers
19 views

Simplification of vectors

Can the expression $(5\vec{u} \times \vec{v})\cdot(2 \vec{u}-7\vec{v})$ be simplified? $\vec{v}$ and $\vec{u}$ are not necessarily part of any orthogonal system. I don't really know how to use any ...
0
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2answers
39 views

Vector Proof for triple product

How can I prove/disprove A x (B x C)=(A x B) x C +B x (A x C) ? I know I could equate the right side to: B(A dot C)-C(A dot B) But I don't know where to go from there.
3
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1answer
29 views

$f(\textbf{x}) = {a\over{|\textbf{x}|^{n-2}}} + b,\text{ }\textbf{x} \neq 0$

If $f(\textbf{x}) = g(r)$, $r = |\textbf{x}|$, and $n \ge 3$, show that$$\nabla^2 f = {{\partial^2 f}\over{\partial x_1^2}} + \dots + {{\partial^2f}\over{\partial x_n^2}} = {{n-1}\over{r}}g'(r) ...
0
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1answer
15 views

Index notation for vector analysis

I have a little question about this step in my solution manual $$ x_j\partial_ka_jx_k - x_j\partial_ka_kx_j = x_ja_j\delta_{kk} -x_ja_k\delta_{kj} $$ the x indices are just from the vector ...
0
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2answers
35 views

Existence of convergent sequences in vector analysis and topology

Many theorems in vector analysis and topology use the concept of converging sequences to e.g. display that a set is closed. E.g. A set F is closed in some metric space M if and only if ...
0
votes
1answer
25 views

What is radius vector? Is it a zero vector?

I only know radius vector is the position vector of any point $(x,y,z)$ in space with respect to origin which in vector notation takes the form $x\vec{i}+y\vec{j}+z\vec{k}$,where ...
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votes
2answers
21 views

parametrization of a curve given by two equations

Find a simple parametrization, $\textbf{r}(u)$, of the curve given by the equations $$4x-y^2=0\\ x^2+y^2-z=0$$ from the point $(0,0,0)$ to $(1,2,5)$ I can't really remember how to do this, even ...
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0answers
25 views

Is it possible to compute the volume of a cone on a inner product space?

This is a matter of curiosity for me. Volumes are often compute using triple integration. But is it possible to compute volumes on a vector space with an inner product defined on that vector space?
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votes
0answers
42 views

How to compute the “jump”, using the divergence theorem,

Given the vector field on $R^3$∖{0}, F(x)=$(x/r^3,y/r^3,z/r^3)$, where $r=(x^2+y^2+z^2)^{\frac {1}{2}}$, let R be a simply connected bounded region, with smooth boundary, in the xy plane containing ...
0
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1answer
59 views

How to show that this function $f$ is continuously differentiable,

Given $f:\mathbb R^2 \to \mathbb R$, and assuming that the directional derivative $D_vf(x)$ is uniformly continuous in $x$ for $|v|=1$, show that $f \in C^1$. This is an old exam question dating back ...
4
votes
3answers
196 views

Is there a simple way of showing that the directional derivative is the dot product of grad(f) with the directional vector u?

I am reading a sort of technical proof but I would like to prove it myself in a cleaner and shorter way. So, I want to show $$D_uf = \nabla f. u$$ Any suggestions are greatly appreciated. ...
0
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1answer
35 views

Do you need subsequences to prove: $A_1, A_2 \subseteq \mathbb{R^n}$ closed sets $\implies$ $A_1 \cup A_2$ closed?

It can be proven that $$A_1, A_2 \subseteq \mathbb{R^n}, \space closed\space sets \implies A_1 \cup A_2 \space is \space closed$$ Using the definition of a closed set in vector spaces. I have a ...
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votes
1answer
27 views

What does it mean for a closed set to contain a sequence?

In vector analysis the closedness of a set can be defined as A set A is closed, iff for each convergent sequence $x_k \in A$, the limit point $a$ of the sequence $x_k$ also belongs to A. What ...
0
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1answer
35 views

Congruent Transformation and Wedge Product

Let $V$ be a three-dimensional Euclidean space. Let $T$ be a congruent transformation over $V$. Let $v$ and $w$ be two vectors in $V$. Is there an interesting relation between $(Tv)\wedge (Tw)$ and ...
4
votes
2answers
200 views

Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator ...
0
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0answers
26 views

Equivalent relation of curves

Given two intervals $I=[a,b],J=[c,d]\subseteq\mathbb{R}$ and two continuous functions $\boldsymbol{u}:I\rightarrow\mathbb{R}^{d}$ and $\boldsymbol{v}% :J\rightarrow\mathbb{R}^{d}$, we say that ...
2
votes
3answers
74 views

What are the dual basis vectors?

What exactly are dual basis vectors such as those which arise in non-orthogonal co-ordinate systems? What is their physical interpretation. Please note, I don't know much tensor calculus yet. I am ...
0
votes
3answers
30 views

How to find the magnitude of a sum of vectors, given the magnitudes of the individual vectors?

I'm at quite a loss as to how to answer this question, and I'd really appreciate some help. The question is as follows: If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that ...
0
votes
2answers
89 views

Solve for the gradient of $\log \sum\limits_{i = 1}^{m} \exp(a_i^Tx + bi)$

This is a standard problem in convex optimization with well known solution but I cannot seem to follow the procedure given in Boyd's CVX book pg 643 Suppose I am given $f(x) = \log \sum\limits_{i = ...
0
votes
0answers
17 views

Equality of line integrals under change of coordinates

Please check my solution to Bamberg & Sternberg, A course in mathematics for students of physics, exercise 7.10(a): Suppose that $u$ and $v$ are curvilinear coordinates on a region $D$ on the ...
7
votes
1answer
64 views

Local coordinates near point such that $X= \partial_1$ is relative to those coordinates, vector field with isolated s.t. coordinates do not exist?

Let $M$ be a finite-dimensional smooth manifold, and let $X$ be a smooth vector field on $M$. Let $X(p) \neq 0$ for some $p \in M$. How do I show that I can find local coordinates near $p$ such that ...