Tagged Questions
2
votes
1answer
62 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 ...
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 ...
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 ...
3
votes
3answers
92 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 $$d=|\mathbf ...
0
votes
2answers
105 views
What does $\nabla \left(\exp{\{\eta^{T}{\bf{u(x)}}\}}{\bf{u(x)}}\right)=\exp{\{\eta^{T}{\bf{u(x)}}\}}{\bf{u(x)}}{\bf{u(x)}}$ mean?
A couple of related questions:
Suppose we want to calculate the gradient $\nabla_{\eta} (\exp{\{\eta^{T}{\bf{u(x)}}\}})$ (as Muphrid suggested, $\nabla_{\eta}$ means the gradient with respect to the ...
22
votes
4answers
505 views
The Meaning of the Fundamental Theorem of Calculus
I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
4
votes
0answers
81 views
vector field as integral
Define a vector field $ \vec{f}(\vec{R}) = \oint_C{|\vec{r} - \vec{R}|^2 d\vec{r} }$ where C is a simple closed curve.
show that there are constant vectors $ \vec{P} $ and $ \vec{Q} $ such that $ ...
1
vote
1answer
71 views
Derivative of the divergence of a vector field with respect to the vector field
Very simply, I am trying to solve the following function in a 3D space:
$$
\frac{\partial}{\partial F}(\nabla\cdot F)
$$
where $F$ is a vector field. I THINK this should be zero based on physical ...
0
votes
1answer
47 views
Use Stokes's Theorem to evaluate the following integral in the easiest way:
$\iint (\bigtriangledown \times V) \cdot n d\sigma $ over the part of the surface $z = 9 - x^2 - 9y^2 $ above the xy-plane, if $V = 2xy\ i + (x^2-2x)\ j -x^2z^2\ k$
My attempt:
The surface is an ...
1
vote
1answer
52 views
Writing $\int_\Omega \Delta u \Delta v$ in a nicer way?
Is there a way to write
$\int_\Omega (\Delta u)^2$ or more generally, $\int_\Omega \Delta u \Delta v$ more nicely (possibly after integrating by parts)? I want something like $\int \nabla f\cdot ...
1
vote
1answer
99 views
Second derivative is what?
I wonder what is the meaning of the second derivative or what kind of object it is when we have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$.
The first derivative is the Jacobian matrix, but ...
1
vote
2answers
147 views
Is the vector field normal or tangential to the curve?
Given the curve $C$, $C = {(x,y):x^2+y^2=1}$, $n=\langle x,y\rangle$ is normal to $C$.
Consider the vector field $F$ defined by $F=\langle y,-x\rangle$.
Is the vector field $F$ tangent to $C$ or ...
0
votes
1answer
150 views
Basic Vector Calculus Problem
This is a basic calculs/pre-calcuus question, that am having trouble with. For real matrices $A_{n \times n}$,$X_{n \times n}$ and $K_{n \times n}$ and a vector $c_{n \times 1}$, I want to have the ...
1
vote
1answer
68 views
Vector Calculus Derivation
I came across the following question in a book I was studying:
Fmagnetic=μ0(M∇)H
Is this the correct expansion below? (I'm not too experienced with vectors operating on the gradient operator)
...
2
votes
1answer
89 views
dot product identity
$$a \cdot (a \cdot b)=(a \cdot a)(a \cdot b)$$
Is this identity true when $a$ and $b$ are vectors, and when $\cdot$ is the dot product operator? And assuming that $()()$ means multiplying the ...
2
votes
1answer
91 views
Why does this equation converge to 1?
The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow ...
1
vote
2answers
191 views
Derivative of cross-product of two vectors
In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
0
votes
1answer
237 views
Find an equation of a plane normal to a given vector
I have to find an equation for a plane normal to the vector $\vec{r(t)} = \langle e^{t}sin(\frac{\pi t}{2}),e^{t}cos(\frac{\pi t}{2}),t^{2}\rangle$ when $t=1$. I know I have to find the derivative and ...
0
votes
1answer
680 views
Finding derivative of dot-product of two vectors
I have to find the derivative of the dot-product of two vectors using the product rule. It took me an hour, checked every component and double checked, and then when I check it on Wolfram, of course ...
1
vote
1answer
2k views
Fleming's “right-hand rule” and cross-product of two vectors
I have been throwing around hand gestures for the past hour in a feeble attempt at trying to solve this question involving a cross product of two vectors $a$ x $b$. So far, I haven't found any ...
2
votes
1answer
599 views
Integration over a triangle
Let $\Delta$ be a triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$ in ${\bf R}^2$.
I want to compute $$I=\int\limits_\Delta x^2\mathrm{e}^{y^2}\;\mathrm{d}A.$$
This is what I've done so far: Note ...
1
vote
2answers
70 views
Continuity and differentiability of a elliptic/paraboloid function
Consider the family of functions $\{f_c \mid c\in\bf{R}\}$ where
$$
\begin{align*}
f_c &\colon{\bf R}^2\to{\bf R}\\
&f_c(x,y)=1-\big(x^2+4y^2\big)^c.
\end{align*}
$$
I intuitively see that ...
4
votes
2answers
2k views
What is the difference between Green's Theorem and Stokes Theorem?
I don't quite understand the difference between Green's Theorem and Stokes Theorem. I know that Green's Theorem is in $\mathbb{R}^2$ and Stokes Theorem is in $\mathbb{R}^3$ and my lecture notes give ...
2
votes
1answer
208 views
Need help computing the partial derivatives of a vector function.
I need help computing the partial derivative shown below. I've never taken a course in vector analysis so I'm not if my previous attempts at solving the problem were even on the right track. If ...
2
votes
3answers
1k views
proving gradient of a scalar field is perpendicular to equipotential surface
let phi = f(x,y,z) be a scalar field, is gradient of phi independent of coordinate choice?
Also how to prove that gradient of phi is perpendicular to scalar potential surface.
0
votes
1answer
200 views
Finding the sign of $\phi$ in spherical coordinates
I know its a little silly, but I got the wrong sign several times.
Just to be clear, $z=r\cos(\phi), -\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$ when converting from cartesian to spherical. So, how do I ...
2
votes
2answers
165 views
Differentiating a function with respect to a vector
I need to differentiate the function $u$ shown below with respect to a vector $\psi$: ($a, c$ and $f$ are constants)
$u(\psi) =\left[\begin{array}{cccc}
a & f & 0 & 0\\
c & a & ...
2
votes
2answers
570 views
Calculating moment of inertia in 2d planar polygon
I've derived equations for a 2D polygon's moment of inertia using Green's Theorem (constant density $\rho$)
$$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} ...
3
votes
4answers
2k views
What's the geometrical interpretation of the magnitude of gradient generally?
In the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal ...
