Read about vector calculus | 105 Discussions | Page 1

  1. Kaguro

    Flux in a rotated cylindrical coordinate system

    ##\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...
  2. Adesh

    I'm not getting the curl of vector potential equal to magnetic field

    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 ...
  3. Adesh

    How to find the curl of a vector field which points in the theta direction?

    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...
  4. Zack K

    Verifying the flux transport theorem

    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...
  5. B

    I Divergence with Chain Rule

    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...
  6. S

    Properties of Four-Vectors

    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...
  7. TheBigDig

    Magnetic field of vector potential

    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}}...
  8. majormuss

    Electrodynamics: Vector Calculus Question

    Why are the red circled Del operators not combining to become 'Del-squared' to cancel out the second term to give a net result of 0?
  9. JD_PM

    Looking for a bunch of solved Sympy problems (Calculus)

    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...
  10. astrocytosis

    Volume integral over a gradient (quantum mechanics)

    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...
  11. JD_PM

    Python Book for learning Python focusing on vector calculus (matrices, eigenvalues, eigenvectors, etc.)

    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.
  12. sams

    I Gauss' Theorem -- Why two different notations are used?

    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...
  13. sams

    I A Question about Unit Vectors of Cylindrical Coordinates

    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}##...
  14. sams

    I Why are central force fields irrotational and conservative?

    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...
  15. sams

    I A question about writing the notation of the nabla operator

    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...
  16. beefbrisket

    I Sign mistake when computing integral with differential forms

    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...
  17. sams

    I Vector Differentiation

    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}...
  18. Xsnac

    Flux of a vector and parametric equation

    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...
  19. H

    Equation for the plane

    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...
  20. C

    Unit Normal to a level surface

    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})...
  21. T

    Find a piecewise smooth parametric curve to the astroid (a hypocycloid with four cusps)

    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) =...
  22. T

    Integration over a ball

    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]...
  23. Mzzed

    Using logarithms in vector Calculus

    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...
  24. Mzzed

    I Method for solving gradient of a vector

    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...
  25. J

    I What is the gradient of a divergence and is it always zero?

    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...
  26. G

    Electric field due to semi-circular wire at a distance

    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...
  27. E

    Programs Vector calculus and E&M physics as a engineering major?

    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...
  28. E

    Line Integral Notation wrt Scalar Value function

    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...
  29. DavideGenoa

    I Laplacian of retarded potential

    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}...
  30. Mateus Buarque

    Calculus Multivariable Calc for IPhO

    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!