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).
1
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0answers
59 views
Vector calculus: Arclength
Please help. I need a guide for this homework question.
Let $u_1 = (x+y)/2$ and $u_2 = (x-y)/2$, where $x$ and $y$ are Cartesian coordinates. Write the differential $d \vec{r}$ as a linear ...
0
votes
2answers
74 views
Is it possible for a scalar (or vector) field to be non-smooth?
Can a 2-dimensional scalar field have a discontinuous contour curve?
How about contour curves that intersect -- possible?
On a related note: can a vector field have a domain that is not defined over ...
1
vote
2answers
143 views
How do I calculate numerically a tensor in polar coordinates?
You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid?
a) fit a (spline) surface, ...
0
votes
1answer
282 views
Find TNB Vectors for a given point
Can anyone tell me whether or not my work and answer below are correct? This is question 13.3.48 in Stewart Calculus 7th edition.
Here is the problem definition:
"Find the vectors $\vec T, \vec N, ...
1
vote
1answer
65 views
Vector Calculus Derivation
I came across the following question in a book I was studying:
Fmagnetic=μ0(M∇)H
Is this the correct expansion below? (I'm not too experienced with vectors operating on the gradient operator)
...
0
votes
1answer
51 views
inequality between entries of the vector and $l_2$ norm of the vector
Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$.
I am wondering for which vectors the following would be true:
$$
\|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n.
$$
Here $c>0$ is ...
3
votes
0answers
51 views
Can I solve for a unique integral kernel?
Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$,
$$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$
Is it possible to solve for the integral kernel, ...
0
votes
2answers
179 views
Multivariable Product Rule, Integration by Parts, Derivative, etc.
I am searching for a book on multi-variable calculus that explains multi-variable product, multi-variable integration by parts, etc.
As an example, here's a simple problem that I would like to be ...
4
votes
2answers
220 views
How to calculate x,y position of 3D points?
I have points in 3D system like this
$$p1=(2,3,4)$$
$$p2=(3,5,5)$$
Here I would like draw point $p1$ and $p2$ in $2D$ view.
Project type = orthographic.
Coordinate system = Cartesian
X- axis, ...
2
votes
1answer
89 views
dot product identity
$$a \cdot (a \cdot b)=(a \cdot a)(a \cdot b)$$
Is this identity true when $a$ and $b$ are vectors, and when $\cdot$ is the dot product operator? And assuming that $()()$ means multiplying the ...
3
votes
1answer
297 views
What does it intuitively mean that the divergence of a vector field is 0?
I was going through an Electrodynamics textbook, and as a prerequisite it requires elemenets of Vector calculus and Multivariable calculus. They discussed divergence, and gave examples of fields with ...
2
votes
0answers
65 views
Changing parametrization with regard to arc length - is it worth it?
If you have found a parametrization $\vec \alpha(s)$ of a curve $C$ for which $\int \lVert \vec \alpha\,'(s)\rVert \mathrm ds$ cannot be expressed in terms of elementary functions, does it make sense ...
1
vote
1answer
222 views
Parametric curve of intersection - line integral with respect to arc length
This comes from Apostol's Calculus, Vol. II, Section 10.9 #14:
A uniform wire has the shape of that portion of the curve of intersecion of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the ...
7
votes
1answer
93 views
What is the average length of all integral curves of a vector field?
Considering a vector field with a source and a sink in a finite comact space, are there any bounds on the length of the integral curves?
Specifically, I am interested in the average length of ...
0
votes
1answer
80 views
Vector analysis can anyone clarify whether my assumptions make sense?
In this problem there are three particles, with velocities $\vec{v_1}, \vec{v_2}$ and $\vec{u}$.
Relative to the particle moving at $u$, the velocities $v_1$ and $v_2$ are of equal magnitude and are ...
0
votes
0answers
71 views
Circle-Circle intersection coordinate system
Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
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0answers
62 views
0
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4answers
75 views
How to get a new point of a vector when rotated.
I want to obtain the new point of a vector that I rotate like this.
When I rotate them, I have the angle of rotation.
I want to know x and y, it rotates taking the reference point of 0,0
Thanks
2
votes
1answer
91 views
Why does this equation converge to 1?
The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow ...
1
vote
1answer
224 views
Chain rule for Hessian matrix
Given $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ smooth and $\phi \in GL(n)$. What is the Hessian matrix $H_{f\circ \phi} = \left(\frac{\partial ^2 (f\circ \phi)}{\partial x_i\partial ...
1
vote
2answers
54 views
Vectorial calculus statement proving
See this statement:
$$
|u×v|^2=|u|^2\cdot|v|^2-(u\cdot v)^2
$$
I need to prove this is right.
I only found that:
$$
u×v=|u|\cdot|v|\cdot\sin\theta
$$
and
$$
u.v=|u|\cdot|v|\cdot\cos\theta
$$
Does ...
0
votes
1answer
105 views
Calculate the normal unit vector for scalar function
In theory, if I have a certain function I can get his normal unit vector by using the gradient of it.
$$\hat{f} = \dfrac{\nabla f}{|| \nabla f ||}$$
Example (correction from answer):
$$ z = 2 -x ...
0
votes
2answers
101 views
Determine the flow and amplitude equation for thermal energy (with Del operator)
It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a $3$-Dimensional space ( e.g $x\cdot y\cdot z = 4\cdot3\cdot2$, a certain room/class of that size ...
2
votes
5answers
309 views
proof for $ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$
this formula just pop up in textbook I'm reading without any explanation
$ (\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{B}\cdot\vec{C})\vec{A}$
I did some "vector ...
0
votes
1answer
172 views
Vector Field of Torus
Explicitly construct a differentiable vector field $W$ in the torus.
Meridians of $T^2$ parameterized by arc length, for all $p \in T^2$, define $W (p)$ as the velocity vector of the meridian ...
2
votes
1answer
71 views
Is this vector derivative correct?
I want to comprehend the derivative of the cost function in linear regression involving Ridge regularization, the equation is:
$$L^{\text{Ridge}}(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T\beta)^2 + ...
0
votes
0answers
45 views
Not sure how to analyze
I am not sure how to do an analysis whereby I have two groups of (7) different sensors that measure the same target over time. As such, my data is simply 7 time series collected at intevals. I want to ...
1
vote
0answers
105 views
Fundamental solution of a vector field
Consider a smooth real vector field $X(x,\partial _x)=\sum a_i(x) \frac{\partial}{\partial x_i}$ on an open subset $\Omega \subset \mathbb{R}^n$ (which is assumed to be sufficiently regular) as a ...
2
votes
2answers
974 views
Product rule for the derivative of a dot product.
I can't find the reason for this simplification, I understand that the dot product of a vector with itself would give the magnitude of that squared, so that explains the v squared. What I don't ...
2
votes
2answers
71 views
Finding direction vector
Can someone please explain how the direction vector was found in problem $2$ of this worksheet?
Below is an image of the problem $2$ of the worksheet.
2
votes
1answer
94 views
How to prove the existence of the following equation?
I learned electrodynamics.
According to the vector potential determination,
$$
\mathbf B = [\nabla \times \mathbf A ],
$$
Coulomb gauge,
$$
\nabla \mathbf A = 0,
$$
and one of Maxwell's equations,
$$
...
1
vote
0answers
113 views
Integrating over a closed contour following vector field
Integrate over a closed contour $c$
$$\oint_c d\vec{r}\times\vec{a}, \quad \vec{a}=-yz\vec{i}+xz\vec{j}+xy\vec{k}$$
where $c$ is cross-section of following two surfaces
$$x^2+y^2+z^2=1$$
and
$$y=x^2$$ ...
0
votes
1answer
25 views
Determine $\alpha$ for which this vector equation takes place.
Assume $ABCD$ is a parallelogram. $O$ is the intersection of the diagonals and $M$ an arbitrary point in the same plan. Determine $\alpha$ for which the following relation takes place:
...
0
votes
0answers
50 views
Finding point distribution by eigen vectors
First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that.
I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
1
vote
1answer
81 views
Simplifying expressions involving real parts
Suppose $\vec{F}(x,y,z)=\vec{f}(x,y)\exp(ikz)$ and $\vec{F}$ satisfies the equations $\nabla \cdot \Re{\vec{F}}=0$ where $\Re{\vec{F}}$ is the real part of $\vec{F}$. It also satisfies $\nabla \times ...
1
vote
1answer
26 views
For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $?
Let $A =(-2, 3, -2)$ and $B =(-6, -1, 1)$. For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $?
I figured, since $P$ is supposed to be on the $x$-axis, the $y$ and $z$ coordinates ...
0
votes
3answers
534 views
Angle of intersection between a line and a plane
I have a line $L$ given by $x = 2 -t$, $y = 1 + t$, $z = 1 + 2t$, which intersects a plane $2x + y - z = 1$ at the point $(1,2,3)$. I have to find the angle which the line makes with the plane. I know ...
1
vote
2answers
189 views
Derivative of cross-product of two vectors
In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
0
votes
1answer
221 views
Find an equation of a plane normal to a given vector
I have to find an equation for a plane normal to the vector $\vec{r(t)} = \langle e^{t}sin(\frac{\pi t}{2}),e^{t}cos(\frac{\pi t}{2}),t^{2}\rangle$ when $t=1$. I know I have to find the derivative and ...
0
votes
1answer
675 views
Finding derivative of dot-product of two vectors
I have to find the derivative of the dot-product of two vectors using the product rule. It took me an hour, checked every component and double checked, and then when I check it on Wolfram, of course ...
0
votes
2answers
135 views
Vector Analysis & Linear Algebra
I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$.
This obviously implies $ ...
3
votes
1answer
103 views
Calculating the divergence of a vector field using non-cubic volumes
The divergence of a vector field is defined is formally defined as:
$\operatorname{div}\,\mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS$
...
2
votes
1answer
4k views
Find the equation of a line which is perpendicular to a given vector and passing through a known point
There is given a vector 2i+j-3k and now I want to find the equation of a line that is perpendicular to the given vector and passing through a known point (1,1,1) .How can I solve this ?.
1
vote
1answer
103 views
Gradient vector function using sum and scalar
Could someone take a look on my attempt to compute the gradient for:
$$f(x) = \lambda \sum_{x = 1}^n g(x_i)$$
Where $x \in \mathbb{R^d}$, $\lambda \in \mathbb{R}$ and
$$g(x_i) = \begin{cases}
x_i - ...
2
votes
3answers
300 views
Confusion regarding orientation of curves in Green's Theorem
I am in the middle of helping some friends out with their vector calculus assignment (I don't do this course). Now in their assignment they have the following question:
Consider the integral ...
1
vote
1answer
95 views
Vector derivative with inner function
I want to compute the gradient for the following function:
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2 + \sum_{j = 1}^k l(\beta_j)$$
where $l(\beta_j) = \begin{cases}
\beta_j - ...
1
vote
0answers
124 views
Minimizing L1 Regularization
I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
1
vote
1answer
39 views
Vector derivative with power of two in it
I want to compute the gradient of the following function with respect to $\beta$
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$
Where $\beta$, $y_i$ and $x_i$ are vectors. The ...
1
vote
1answer
71 views
basic vector being hermitian
If the space has a mixed metric signature, not all the basis vectors are Hermitian.
Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose
conjugate is, ...
1
vote
2answers
160 views
Log-likelihood gradient and Hessian
Considering a binary classification problem with data $D = \{(x_i,y_i)\}_{i=1}^n$, $x_i \in \mathbb{R}^d$ and $y_i \in \{0,1\}$. Given the following definitions:
$f(x) = x^T \beta$
$p(x) = ...
