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).
0
votes
1answer
24 views
Wedge product of vector fields
Can somebody explain me step by step how can I compute the wedge product $X\wedge Y$ of two vector fields, $X,Y$, in $\mathbb{C}^2$?
We can consider
$$
X=X_1\partial_x+X_2\partial_y
$$
and
$$
...
2
votes
2answers
49 views
Can flux be proportional to $r^2$ in divergence theorem?
The motivation for the divergence being interpreted as the flux of stuff used the following:
$$\text{div} F(a) = \lim_{r\to0}\frac{3}{4\pi r^3}\int_{|x-a|=r} F\cdot n dA$$
Without the $r^3$ in the ...
1
vote
2answers
38 views
Vectors: Finding third point in a right angle triangle
Given A(1,4) and B (3,-5) use the dot product to find point C so that triangle ABC is a right angle triangle.
0
votes
1answer
38 views
Which of the following integrals is =0?
$\int_C \bar z dz$ where the curve $C$ is described by $|z|=1$
so the integral of $\bar z=x-iy$, why does it equal zero?
2
votes
1answer
60 views
Trouble understanding a common vector calculus example
I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
2
votes
1answer
45 views
Divergence free, smooth functions on unit circle.
I need to construct a divergence free, smooth, vector function on a unit circle such that
$ \mathbf{u} = (u_1,u_2) = (0,0) $ on $\partial B$, and $\int\limits_B u_i \neq 0, \ i=1,2$. I was able to ...
1
vote
1answer
24 views
How meaningful on an unique $(c_1\nabla b_1+c_2\nabla b_2+c_3\nabla b_3)$??
Since the few (smooth) vector fields can denote themself in the unit vectors,
$$\boldsymbol{\dbinom{B}{C}}=\dbinom{b_1\boldsymbol i+b_2\boldsymbol j+b_3\boldsymbol k}{c_1\boldsymbol i+c_2\boldsymbol ...
3
votes
1answer
44 views
Fourier transform of gradient
I encountered in a physics book the Fourier transform $F$ of the gradient of a function $g$ smooth with compact support on $\mathbb R^3$. Up to some multiplicative constants:
$F(\nabla ...
2
votes
1answer
50 views
No ideas to collapse $\boldsymbol{(\nabla\times B)\times C-(B\times\nabla)\times C}+\boldsymbol\nabla(\boldsymbol B\bullet\boldsymbol C)$
I v expanded the vector calculus terms and added them :)
...
2
votes
0answers
38 views
Notation of a vector field
Usually vector field looks like $\sum_{i} a_i(x_1,x_2,...x_n) \frac{\partial}{\partial x_i}$, also it is not a problem to write it like that $\sum_{i}a_i\partial_i$ or even $a^i\partial_i$ using ...
1
vote
1answer
39 views
Help with Gradient-related concepts
I'm trying to understand the concept of a Gradient vector, and it seems I'm having trouble visualizing certain stuff. So, I was hoping if someone could resolve some of the questions I'm having on my ...
0
votes
0answers
13 views
Is there a formula to get the changes in ship course from wind and current?
Anyone know how to get the changes of degree's in ship course that affected by wind and current?
I thinks it maybe related with the speed and degree of WIND and CURRENT. But I don't know how to ...
1
vote
1answer
27 views
Collinearity of three points of vectors
Show that the three vectors $$A\_ = 2i + j - 3k , B\_ = i - 4k , C\_ = 4i + 3j -k$$ are linearly dependent. Determine a relation between them and hence show that the terminal points are collinear.
...
4
votes
1answer
74 views
Multivariable Calculus Integral Proof
This problem is being very difficult for me to solve, I need help.
Consider $F:\mathbb{R}^2\rightarrow\mathbb{R}$ of class $C^1$, suppose that the level curves of $F$ are closed and that $\nabla F$ ...
3
votes
0answers
63 views
Please help tell me what i am doing wrong for multivariable calculus problem [duplicate]
Suppose $F =(2x−4y)i +(x+3y)j$. Use Stokes' Theorem to make the following circulation calculations:
(a) Find the circulation of $F$ around the circle $C$ of radius $10$ centered at the origin ...
2
votes
1answer
90 views
Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory
In order to proof the following identity:
$$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$
Instead of checking this by brute force, Landau writes de product of ...
1
vote
1answer
46 views
Let $F(x,y,z) = -c(r/||r||^3)$ be the force resulting from the inverse square law…
$c$ is a constant and $r = (x,y,z)$. Show that $\displaystyle f(x,y,z) = \frac{c}{\sqrt{x^2+y^2+z^2}}$ is a potential function for $F$. What can be concluded from any path from point $A$ to point $B$ ...
0
votes
2answers
53 views
Stokes Theorem Integral
Evaluate $\iint_S \langle F,\eta\rangle \,d\sigma$ where $F(x,y,z)=(xz,yz,z^2)$ and $S$ is the upper hemisphere of radius $1$ centred at the origin.
$\eta$ is the unit vector perpendicular to the ...
0
votes
0answers
27 views
show that r perpendicular to F for vectors r & F at any point
First I sketched the vector field F by sketching points in each quadrant.
Then I sketched r from points that were given. How can I show that r is perpendicular to F for vectors r & F at any ...
0
votes
0answers
28 views
left handed and right handed cartesian coordinates ?
during my reading in Vector analysis - Edwin wilson - when i reached to the part of "unit vectors i , j , k" page 20 he stated a introduction of solid cartesian coordinates stating that regular ...
1
vote
2answers
25 views
Vector orthogonal to U
Let y=(3,5) and u=(6,2).
Write y as the sum of a vector in Span{u} and a vector orthogonal to u.
If someone could do this problem as an example, it would be great.
1
vote
3answers
63 views
Interpreting the Surface Integral over a Vector Field
I have seen the fact that in certain instances, the Surface Integral over a Vector Field gives the quantity of fluid flowing through the surface in unit time (as in here, or in any standard Vector ...
2
votes
1answer
56 views
Convex cone question.
Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
0
votes
2answers
71 views
Question regardng Stokes' theorem
I have a very simple question. The Calculus book I am using provides this question (along with a solution) as an exercise in Stokes' theorem.
First of all, I have no idea why they have a picture of a ...
1
vote
1answer
59 views
Gradient and curl operators
I have some troubles with vector identities for the gradient and curl operators, for example something like the gradient of the vector or the cross product :
Vector calculus identities
since i have ...
0
votes
1answer
37 views
Show that anecessary and sufficient condition for $x_{p}$ to be tangent to $S^{n}$ at $p$
Please help me! How do I solve this problem? I didnt produce any idea because I didnt understand this topic properly. Thus, please can you explain the solution explicitly? Thank you for help:)
1
vote
1answer
55 views
Directional derivative of a scalar field in the direction of fastest increase of another such field
Suppose $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are scalar fields. What expression represents the directional derivative of $f$ in the direction in which $g$ is increasing the fastest?
8
votes
4answers
414 views
Why vector calculus seems inconsistent and vague
I am a senior student of engineering and I have been studying calculus for a while when I reached the part of vector calculus I felt that this part is inconsistent and there is a multiple questions ...
0
votes
0answers
64 views
Gradient contradicting dimensions. Find the mistake!
$\nabla diag(X^TX)= diag(\nabla(X^TX))=2diag(X)?$ ?- when $X$ is non symmetric rectangular matrix with real entries. $diag(.)$ denotes a diagonal matrix formed with the diagonal elements being the ...
0
votes
2answers
35 views
Gradient of scalar potential
Say we have scalar potential in a form
$$
U = A \ln (\vec{a} \times \vec{r})^2 e^{-\vec{b} \cdot \vec{r}}.
$$
How would one calculate gradient $\vec{E}=-\nabla U$ of such potential?
A is a ...
0
votes
1answer
70 views
How to find $\operatorname{div} F$ and $\operatorname{curl} F$ of the vector field $F=\hat r=\cos\theta \hat{\imath} + \sin\theta \hat{\jmath}$
I was given a bunch of divergence and curl questions in class but I am stumped on this one. If anyone can help explain what I should do with it I would appreciate it.
Calculate $\operatorname{div} ...
1
vote
1answer
49 views
Different geometrical concepts of vectors
I'm a bit confused about the various geometric concepts of vectors.
I'm mainly trying to understand if we can classify any vector into one of two categories.The first category would be free ...
2
votes
2answers
30 views
Find all values of $a$ such that $w = ai- \frac{a}{3}j$ is a unit vector
Find all values of $a$ such that $w = ai- \frac{a}{3}j$ is a unit vector.
I just need help understanding precisely what this question is asking. This is the first chapter and section of multivariable ...
0
votes
2answers
63 views
Whats the connection between functions with curl 0 and holomorphic functions
When I first saw the Cauchy-Riemann differential equations they remind me on the conditions for the curl of a function to be zero.
Here some notation I will use:
$$\frac{\partial f}{\partial x} = ...
0
votes
3answers
93 views
Geometrical Proof of a Rotation
I wanna prove geometrically ( and not by linear algebra, doing transformations in the bases ) the result of the rotation of a point. The proof should only include geometrical steps like using ...
1
vote
5answers
75 views
Can I treat vectors in $\Bbb R^3$ with a component equal to zero as vectors in $\Bbb R^2$?
If I have a vector $(0,4,4)$ and was finding perpendicular unit vectors to this, is it the same case as finding perpendicular unit vectors for the vector $(4,4)$? Meaning a maximum of two possible ...
0
votes
2answers
36 views
Prove that 3 points are not on the same line
Given $P_1=(1, 1, 1)$, $P_2=(2, -1, 2)$ and $P_3=(3,0,1)$, I need to prove that these three points are not on the same line.
What I tried - I showed that $\vec{P_1P_2}$, $\vec{P_1P_3}$ and ...
0
votes
0answers
32 views
Question about vector fields
Let $\vec{r} = x \vec{i} + y \vec{j} + z \vec{k}$ and $\vec{a} = 5 \vec{i} + 6\vec{j} + \vec{k}$.
Find $\nabla(\vec{r} \cdot \vec{a})$.
Let $C$ be a path from the origin to the point ...
3
votes
3answers
91 views
The distance between a pair of skew diagonals on two adjacent faces of a cube.
Say we're interested in the distance between the diagonals $u=(0,0,0)+(1,0,1)t$ and $v=(0,0,1)+(1,1,0)s$ of a unit cube.
The standard formula for the distance between two skew lines $$d=|\mathbf ...
0
votes
2answers
53 views
Integral of Vector Valued Function
I'm in doubt in how to precisely define the integral of a vector valued function. And here, I'm not saying: integral of the dot product of two vectors, but integral of the vector itself. For instance, ...
0
votes
0answers
42 views
Optmizing sum of two vectors
I apologize in advance for the title, but I don't know how to express exactly what I want to do.
So, here's my problem: I have 66 vectors, each one with 8 values, those values can be positive or ...
2
votes
1answer
87 views
Proving that the value of the integral doesn't depend on the surface
I'm trying to prove that if we have the vector field $v : \mathbb{R}^n \to \mathbb{R}^n$ given in spherical coordinates by:
$$v(\rho, \theta, \phi)=\frac{1}{\rho^2}\hat{\rho}$$
Where $\hat{\rho}$ is ...
0
votes
3answers
57 views
Solving a differential equation with more than one dependent variable
It's been awhile since I took differential equations. Now I am using differential equations in another class. This is why you shouldn't sell back books from your major courses. :)
How would I solve ...
0
votes
2answers
105 views
What does $\nabla \left(\exp{\{\eta^{T}{\bf{u(x)}}\}}{\bf{u(x)}}\right)=\exp{\{\eta^{T}{\bf{u(x)}}\}}{\bf{u(x)}}{\bf{u(x)}}$ mean?
A couple of related questions:
Suppose we want to calculate the gradient $\nabla_{\eta} (\exp{\{\eta^{T}{\bf{u(x)}}\}})$ (as Muphrid suggested, $\nabla_{\eta}$ means the gradient with respect to the ...
0
votes
0answers
22 views
A vector map $f$ with the $f(k_1, .., ..k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1+k_{n+1}, …, k_n + k_{2n})$
Is it possible to have a vector map $f: V^n \rightarrow \mathbb{R}$ with the $f(k_1, .., ..k_n) + f(k_{n+1}, .., k_{2n}) = f(k_1+k_{n+1}, ..., k_n + k_{2n})$? If so, is there a name given to this type ...
21
votes
4answers
499 views
The Meaning of the Fundamental Theorem of Calculus
I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
1
vote
1answer
199 views
Moving point along the vector [closed]
I'm making a game. I have came across a problem. I have to move a point along a vector for some distance. Can anyone help me? Any ideas?
1
vote
0answers
21 views
vector analysis and reduce to a general expression in theta's
$\vec{A},\vec{B},\vec{C}$ and $\vec{D}$ are unit vectors ($|A|=1,|B|=1,|C|=1$ and $|D|=1$). The angle between the vectors,
1) $\vec{A}$ and $\vec{B}$ is $\theta_{1}$ ...
4
votes
0answers
81 views
vector field as integral
Define a vector field $ \vec{f}(\vec{R}) = \oint_C{|\vec{r} - \vec{R}|^2 d\vec{r} }$ where C is a simple closed curve.
show that there are constant vectors $ \vec{P} $ and $ \vec{Q} $ such that $ ...
2
votes
2answers
90 views
Is there the shortest notation for a vector obtained by projecting $\vec{A}$ onto $\vec{B}$?
Is there the shortest notation defined for a vector obtained by projecting
$\vec{A}$ onto $\vec{B}$?
Is there the shortest notation defined for the complementary vector of a vector obtained by ...
