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).
9
votes
3answers
339 views
How is “area” a vector?
"We consider Area as a vector."
How is an area a vector? Why is that the vector is always normal to the area element?
1
vote
1answer
56 views
eigen value of the gradient operator
Eigen value of the following differential equation
$$\nabla \phi (\vec r) = a \vec {k} \phi(\vec{r})$$
is
$$ \phi(\vec{r}) = e^{a \vec{k}.\vec{r}}$$
How can i derive this result?
5
votes
3answers
134 views
Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.
Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
1
vote
1answer
52 views
Writing $\int_\Omega \Delta u \Delta v$ in a nicer way?
Is there a way to write
$\int_\Omega (\Delta u)^2$ or more generally, $\int_\Omega \Delta u \Delta v$ more nicely (possibly after integrating by parts)? I want something like $\int \nabla f\cdot ...
0
votes
1answer
77 views
Curvature $\kappa$ Proof
The curvature of a curve (rate of change of the unit tangent vector with respect to arc length) is defined as $$\kappa = \frac{|\underline{r}'(t) \times \underline{r}''(t)|}{|\underline{r}'(t)|^3}$$ ...
0
votes
2answers
160 views
How to show that $(r\times\nabla)\cdot(r\times\nabla)=r\cdot[\nabla\times(r\times\nabla)]$?
A friend asked how to show that
$$(r\times\nabla)\cdot(r\times\nabla)=r\cdot[\nabla\times(r\times\nabla)]$$
$r$ is a position vector, $\nabla$ is the grad operator, and $\cdot$ and $\times$ are the ...
1
vote
1answer
98 views
Second derivative is what?
I wonder what is the meaning of the second derivative or what kind of object it is when we have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$.
The first derivative is the Jacobian matrix, but ...
0
votes
1answer
48 views
Surface Integral Problem and Formula
For this question:
Calculate $\int \!\!\! \int E\cdot \vec n d\sigma$ where S is the parametric surface $X(s,t)=[st,s^2,t^2]^T$, $0\le s\le t\le 1$, and the E is the vector field ...
3
votes
0answers
154 views
Stokes' and Divergence Theorem Problems
I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly.
Question 1
Let C be the boundary of the surface ...
3
votes
1answer
56 views
Vector Equation spelled out
I am looking to find the $ x_1, y_1 $ coordinate that is distance $d$ from a starting point of $x_0, y_0$ coordinate as traveled along a line ($y=mx+b$)
I was going to ask this question here, but I ...
0
votes
1answer
38 views
How to explain the solution
How to write an equation for plane, which includes dots with radius-vectors $\mathbf r_{1}, \mathbf r_{2}, \mathbf r_{3}$ that do not lie on a straight line?
The answer is
$$
(\mathbf r, ([\mathbf ...
2
votes
1answer
135 views
Proof of $\nabla\times\vec{E}=0\Rightarrow\exists\Phi:\vec{E}=\nabla\Phi$
Would anyone be able to help me prove these two statements involving vector fields?
For a suitable region
$$\nabla\times\vec{E}=0\Rightarrow\exists\Phi:\vec{E}=\nabla\Phi$$
and
...
2
votes
0answers
46 views
Dot products of three or more vectors
Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ ...
2
votes
1answer
57 views
laplacian of a function, relation to the function
Suppose for some function $\Phi$ we have:
$$
\nabla^2 \Phi(\mathbf{r})=\phi(\mathbf{r})
$$
where $\phi(\mathbf{r})$ is some well-behaved smooth function, which is finite everywhere.
Does this mean ...
0
votes
1answer
138 views
making three parallel lines (3d) with equal distance seperation
I have three parallel lines (3d lines). say AB, CD, EF. The center line i.e. CD is given by intersecting the two planes by which the AB, DE lie on. The shortest distance between AB and CD (say d1) is ...
2
votes
1answer
246 views
Vector Triple Product with Nabla Operator
I want to prove $$\vec{A}\times(\nabla\times\vec{A})=\frac{1}{2}\nabla(A \cdot A)-(\vec{A}\cdot\nabla)\vec{A}$$
This looks strikingly similar to the BAC CAB formula
...
0
votes
1answer
68 views
compute overlapping % of 2 parallel lines
I have two parallel line segments, say AB, CD. If I project the end
points onto a common third parallel line, then I want to know the
portion of overlap made by above 2 lines. I think I should ...
1
vote
2answers
147 views
Is the vector field normal or tangential to the curve?
Given the curve $C$, $C = {(x,y):x^2+y^2=1}$, $n=\langle x,y\rangle$ is normal to $C$.
Consider the vector field $F$ defined by $F=\langle y,-x\rangle$.
Is the vector field $F$ tangent to $C$ or ...
1
vote
3answers
103 views
About $f(x,y)=\frac{1}{1+|x|+|y|}$ integrability
For what values of $1\le p \le \infty$ does $f(x,y)=\frac{1}{1+|x|+|y|}$ with $(x,y) \in \mathbb{R}^2$ belong to $L^p(\mathbb{R}^2)$?
Using Wolfram Alpha I've found that the answer should be $p > ...
0
votes
2answers
137 views
Needing an example of one riemann integrable function
This is easy, but I couldn't find some example of a function that is not integrable but its Riemann improper integral exists and is finite
1
vote
1answer
132 views
About the measurable subsets and the Lipschitz condition
I have, again, a doubt with the measurable subsets.
If I have that $T\colon\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is Lipschitz, does $T$ send Lebesgue measurable sets in Lebesgue measurable sets. ...
1
vote
1answer
63 views
About the properties of Lebesgue measurable subsets
This is a doubt about Lebesgue measurable subsets
If I have two Lebesgue measurable subsets $E_1, E_2$ in $\mathbb{R}$, is the subset $E_1\times E_2$ Lebesgue measurable in $\mathbb{R}^2$?, If it ...
1
vote
0answers
65 views
About Lebesgue measure
This is a problem of Lebesgue measure and measure theory specifically.
Suppose that
$f:\mathbb{R}^2\longrightarrow [0,\infty)$ is measurable.
$\Omega_1\subseteq \mathbb{R}^2$ is Lebesgue ...
2
votes
2answers
93 views
Question about n-forms and $d^2 = 0$
In my late question the answer was great, but I didn't understand what is the 0-form, 1-form and 2-form. Are this can be genralized for any natural $n$?
they are both special cases of the fact ...
0
votes
1answer
44 views
$\nabla \times X$ is given on a surface, can I show that X = 0 on the same surface
If I have a volume V enclosed by a surface S, and $\nabla \times X$ is given on the surface, what information does that give me about X on S. Is there a method of showing that X = 0 on S? (in the ...
1
vote
1answer
160 views
Prove the functions are unique in a volume, vector calculus problem
I am working through the following problem, but finding it hard to know where to go.
Using the Divergence theorem and the following identities
$\nabla .(A \times B) = B.(\nabla \times A) - A.(\nabla ...
1
vote
1answer
10 views
change in t(x,y,z) in the direction (a,b) on f(x,y)
Given a surface $z=f(x,y)$ we need to find the change in temperature $t(x,y,z)$ in the direction of $(a,b)$ at point $(x_0,y_0)$.
My current way of thinking is finding the tangent plane of $f(x,y)$ ...
12
votes
3answers
384 views
Does taking $\nabla\times$ infinity times from an arbitrary vector exists?
Is it possible to get the value of:
\begin{equation}
\underbrace{\left[\nabla\times\left[\nabla\times\left[\ldots\nabla\times\right.\right.\right.}_{\infty\text{-times taking curl ...
3
votes
2answers
272 views
Does the Implicit mapping theorem imply the inverse mapping theorem?
Does the Implicit mapping theorem imply the inverse mapping theorem?
1
vote
2answers
54 views
About regular surfaces
I never had seen this exercise, but I'm confused again, I don't know what I have to use.
I have the surface $S=\{(x,y,z)\in \mathbb{R}^3|xy+xz+yz=1,x>0,y>0,z>0\}$, is $S$ regular?. Then, if ...
0
votes
1answer
81 views
The implicit and the inverse function
This is a simple problem but I am confused about the results.
Suppose the $f:\mathbb{R}^2\longrightarrow\mathbb{R}^2$ is a differentiable mapping in $\mathbb{R}^2$ such that $\det(d_pf)\neq 0$ for ...
4
votes
1answer
87 views
physical meaning of the vector $(A \cdot \nabla) A$
If we consider the vector $\left ( A \cdot \nabla \right) \: B$, we have in Cartesian coordinates
$$\left ( A \cdot \nabla \right) \: B = \left ( A \cdot \nabla B_x \right ) e_x + \left ( A \cdot ...
1
vote
2answers
182 views
Equation of a line on a plane…
Hi this question belongs to camera projections but i cannot understand the mathematics...
i am not getting how the cross product of two vectors (underlined in red) gives the equation of a ...
-2
votes
1answer
91 views
Find if possible an orthogonal unit vector at: 2i + 3j - k and - 2i - 3j + 4k
The question is:
Find, if possible, an orthogonal unit vector at: $2i + 3j - k$ and $-2i - 3j + 4k$.
$$\left|\begin{matrix} i & j & k \\ 2 & 3 & -1 \\ -2 & -3 & 4 ...
0
votes
1answer
101 views
How would one use matrices to find a normal unit vector?
A recent class assignment involved finding a unit vector perpendicular to a plane, given two unit vectors to start with. The solution given involved using the cross product; I was wondering if such a ...
1
vote
1answer
117 views
proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$
I encounter this triple product property in wikipedia
But I can't find proof for
$$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$
The RHS cross ...
1
vote
0answers
56 views
Flow in a vector field
It's easy, but I don't know how to calcule the flow $\iint_S{\bf F}\cdot d{\bf S}$ of the Field ${\bf F}(x,y,z)=\langle y^2,xz,2z\rangle$ through de surface $z=1-x^2-y^2$, $z\geq 0$ whose normal ...
4
votes
1answer
43 views
Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature
So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve ...
0
votes
2answers
271 views
Find the directions in which the directional derivative has the value 1
Can anyone show me how to adjust my work below so that it is a correct answer? This is question number 14.6.28 in the 7th edition of Stewart Calculus.
Find the directions in which the directional ...
0
votes
1answer
150 views
Basic Vector Calculus Problem
This is a basic calculs/pre-calcuus question, that am having trouble with. For real matrices $A_{n \times n}$,$X_{n \times n}$ and $K_{n \times n}$ and a vector $c_{n \times 1}$, I want to have the ...
0
votes
1answer
67 views
Calculate the following contour integral.
Using Stokes theorem
I seem to find the integral value as $2\pi r^2$. Can anyone help me if I am right?
How do i do it without using Stokes theorem?
$$C=\oint\limits_K \mathrm d \mathbf r\cdot ...
0
votes
1answer
42 views
Derivative of a vector function
Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus.
Find the derivative of the vector function:
...
1
vote
2answers
85 views
Calculating the following integral .
I have found the following integral to be zero, but i don't think its correct.
$$
C = \oint_K d\mathbf r\cdot \mathbf A
$$
Where $\mathbf A = \frac 1 2 \mathbf n \times \mathbf r$ and $\mathbf n \cdot ...
1
vote
2answers
34 views
How are vectors defined in terms of sequences?
I'm reading this Wikipedia article on sequences Sequence where it mentions 'Sequences over a field may also be viewed as vectors in a vector space.' under vectors section. I'm not able to grasp this, ...
0
votes
1answer
41 views
problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
can anyone expand on how the final answer is derived?
I try to expand but ended up scratching my head.
the best I can ...
1
vote
1answer
25 views
what is a vector of polyhedron?
What does v mean in the following, does it a point inside the polyhedron?
2
votes
1answer
616 views
Proving vector calculus identity $\nabla \times (\mathbf a\times \mathbf b) =\cdots$ using Levi-Civita symbol
I want to prove the following relation
$$\nabla \times (\mathbf a\times \mathbf b) = \mathbf a\nabla \cdot \mathbf b + \mathbf b \cdot \nabla \mathbf a - \mathbf b \nabla \cdot \mathbf a - ...
2
votes
2answers
127 views
How to prove the equality of two vectors?
OK, i am trying to prove that if $\vec a\times \vec b = \vec a \times \vec c$
and also $\vec a\cdot \vec b = \vec a \cdot \vec c$ then $\vec b = \vec c$.
so far i got to $\vec n \tan \alpha = \vec m ...
3
votes
1answer
125 views
What is the name of this equation?
I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture?
I have also tried to re copy it:
$$
...
0
votes
0answers
38 views
Knowing $\alpha$ and $\beta$, compute $\gamma$. $vers(\vec v)=(\cos\alpha,\cos\beta,\cos\gamma)$
Knowing that:
$\vec v=|\vec v| vers(\vec v)$
$\cos\alpha=\frac{\vec v \cdot \vec i}{|\vec v|\cdot|\vec i|}$
$\cos\beta=\frac{\vec v \cdot \vec j}{|\vec v|\cdot|\vec j|}$
$\cos\gamma=\frac{\vec v ...
