##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ##
## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume...
In this image of Introduction to Electrodynamics by Griffiths
.
we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...
I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$
(##r## is the distance from origin, ##\phi## is azimuthal angle and...
Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)##
I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see
\frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2
K is a constant and T...
Two four-vectors have the property that ##A^\mu B_\mu = 0##
(a) Suppose ##A^\mu A_\mu > 0##. Show that ##B^\mu B_\mu \leq 0##
(b) Suppose ##A^\mu A_\mu = 0##. Show that ##B^\mu## is either proportional to ##A^\mu## (that is, ##B^\mu = k A^\mu##) or else ##B^\mu B_\mu < 0##.
Part (a) is...
So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following
\frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}...
Two weeks ago I had no idea on how to code using Python. Now I have completed an online course on functions, loops and strings. However, in that course I did not practice using the specific library called Sympy. Besides, I will use Python in the Physics-Math background, for solving problems like...
Homework Statement
1) Calculate the density of states for a free particle in a three dimensional box of linear size L.
2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0##
3) Calculate the integral ##\int...
I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus).
If you also have online recommendations please feel free to write them.
In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as:
In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as:
Kindly I would like to know please:
1. What is the difference between...
I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...
In Mathematical Methods for Physicists, 6th Edition, page 44, Example 1.8.2, the curl of the central force field is zero.
1. Why are central force fields irrotational?
2. Why are central force fields conservative?
Any help is much appreciated...
I have a simple question about the notation of the nabla operator in Vector Analysis. The nabla operator is a vector differential operator and it is written as:
$$\nabla = \hat{x} \frac {∂} {∂x} + \hat{y} \frac {∂} {∂y} + \hat{z} \frac {∂} {∂z}$$
Is it okay if we accented nabla by a right...
The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
I have a question regarding the dot product and the cross product differentiation. I was wondering whether:
$$\frac{d(\vec{A}.\vec{B})}{du} = \vec{A}. \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} .\vec{B}$$
is the same as
$$\frac{d(\vec{A}.\vec{B})}{du} = \frac{d\vec{A}}{du} .\vec{B} + \vec{A}...
Homework Statement
Compute the flux of a vector field ##\vec{v}## through the unit sphere, where
$$ \vec{v} = 3xy i + x z^2 j + y^3 k $$
Homework Equations
Gauss Law:
$$ \int (\nabla \cdot \vec{B}) dV = \int \vec{B} \cdot d\vec{a}$$
The Attempt at a Solution
Ok so after applying Gauss Law...
Homework Statement
Problem: Please find an equation for the plane that contains the point <3, -2, 4> and that includes the line given by (x-3)/2 = (y+1)/-1, z=5 (in symmetric form). Simplify
Homework Equations
I'm really not sure where to start and what process to take to arrive to my answer...
Homework Statement
Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$
Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0)
Homework Equations
$$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$
$$ z = \sqrt{x^{2}+y^{2} -1} $$
$$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...
Homework Statement
Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve?
Homework Equations
$\phi(\theta) =...
Homework Statement
I'm working on a generalization of gravitation to n dimensions. I'm trying to compute gravitational attraction experienced by a point mass y due to a uniform mass distribution throughout a ball of radius a -- B(0, a).
Homework Equations
3. The Attempt at a Solution [/B]...
Homework Statement
My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just cancelled the relevant n terms and then identified 1/n as the derivative of L however if this were...
I have seen two main different methods for finding the gradient of a vector from various websites but I'm not sure which one I should use or if the two are equivalent...
The first method involves multiplying the gradient vector (del) by the vector in question to form a matrix. I believe the...
Hi Folks,
Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...
Homework Statement
A semi-circular wire containing a total charge Q which is uniformly distribute over the wire in the x-y plane. the semi-circle has a radius a and the origin is the center of the circle.
Now I want to calculate the electric field at a point located on at distance h on the...
I am an engineering major at Los Angeles Pierce community college. I have been for the last years working 40 hours a week in order to sustain and put myself through community college. After I transfer, I don't plan on working. Now, each semester due to my work schedule and life happening, I can...
I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed.
Homework Statement
Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl##
along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.
Homework...
Dear friends,
I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by
$$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
Hi guys, i´m pretty well in calculus 1 and i´m studying for the International Physics Olympiad. So I´d like to know some multivariable calculus books that cover vector calc too, are balanced (proofs are welcome) and emphasizes physical intuitions. Thank you already!