Read about vector calculus | 106 Discussions | Page 2

  1. 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!
  2. M

    Vector Analysis Problem Involving Divergence

    Homework Statement [/B] Let f and g be scalar functions of position. Show that: \nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f) Can be written as the divergence of some vector function given in terms of f and g. Homework Equations [/B] All the identities given at...
  3. J

    Finding the curl of velocity in spherical coordinates

    Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...
  4. Angelo Cirino

    I Laplacian in integration by parts in Jackson

    I am reviewing Jackson's "Classical Electromagnetism" and it seems that I need to review vector calculus too. In section 1.11 the equation ##W=-\frac{\epsilon_0}{2}\int \Phi\mathbf \nabla^2\Phi d^3x## through an integration by parts leads to equation 1.54 ##W=\frac{\epsilon_0}{2}\int |\mathbf...
  5. C

    I Proofs of Stokes Theorem without Differential Forms

    Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...
  6. maxhersch

    I Kronecker Delta and Gradient Operator

    I am looking at an explanation of the gradient operator acting on a scalar function ## \phi ##. This is what is written: In the steps 1.112 and 1.113 it is written that ## \frac {\partial x'_k} {\partial x'_i} ## is equivalent to the Kronecker delta. It makes sense to me that if i=k, then...
  7. M

    Vector Calculus - Tensor Identity Problem

    Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...
  8. toforfiltum

    Evaluating Cartesian integral in polar coordinates

    Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework Equations The Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0...
  9. toforfiltum

    Find center of mass of planar quadrilateral

    Homework Statement Consider the planar quadrilateral with vertices (0, 0), (2, 0), (1, 1) and (0, 1). Suppose that it has constant density. What is its center of mass? Homework Equations The Attempt at a Solution Since it has constant density, could I assume that the center of mass would be...
  10. toforfiltum

    Finding transformation T such that T(D*)=D

    Homework Statement If ##D^*## is the parallelogram whose vertices are ##(0,0)##,##(-1,3)##, ##(1,2)##, and ##(0,5)## and D is the parallelogram whose vertices are ##(0,0)##, ##(3,2)##,##(1,-1)## and ##(4,1)##, find a transformation ##T## such that ##T(D^*)=D##. Homework Equations The Attempt...
  11. toforfiltum

    Find ratio between dimensions of can w/ largest capacity

    Homework Statement A cylindrical metal can is to be manufactured from a fixed amount of sheet metal. Use the method of Lagrange multipliers to determine the ratio between the dimensions of the can with the largest capacity. Homework Equations The Attempt at a Solution $$V(r,h)=\pi r^2h$$ $$2...
  12. toforfiltum

    Describe curve to reduce field intensity in fastest time

    Homework Statement Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point ##(x,y)## is given by ##M(x,y)=3x^2+y^2+5000##. If Igor is at the point ##(8,6)##, describe the curve along which he should travel if he wishes to reduce the field...
  13. M

    I What will be the 4th axis of a 3d curl?

    Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x axis and similarly x-z plane rotating wrt to y axis. My question is...
  14. toforfiltum

    Newton's method to approximate critical point

    Homework Statement Let ##F(x,y)=4sin(xy)+x^3+y^3## Use Newton's method to approximate the critical point that lies near ##(x,y)=(-1,-1)## Homework Equations The Attempt at a Solution I have a problem here because the derivative is not a square matrix. Hence, I can't find the inverse needed...
  15. toforfiltum

    Conflicting result in derivative of composite function

    Homework Statement Let $$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2} \space & \text{if} \space(x,y)\neq(0,0)\\0 \space & \text{if} \space(x,y)=(0,0)\end{cases}$$ a) Use the definition of the partial derivative to find ##f_x(0,0)## and ##f_y(0,0)##. b) Let a be a nonzero constant and let...
  16. toforfiltum

    Continuity of ##g(x,y)## and its partials

    Homework Statement Let ##g(x,y)=\sqrt[3]{xy}## a) Is ##g## continuous at ##(0,0)##? b) Calculate ##\frac {\partial g}{\partial x}## and ##\frac{\partial g}{\partial y}## when ##xy \neq 0## c) Show that ##g_{x}(0,0)## and ##g_{y}(0,0)## exist by supplying values for them. d) Are ##\frac...
  17. toforfiltum

    Find partials for ##f## and the equation of plane

    Homework Statement Function is ##f(x,y)=((x-1)y)^\frac{2}{3}##,##\space\space(a,b)=(1,0)## a) Calculate ##f_{x}(a,b)## and ##f_{y}(a,b)## at point ##(a,b)## and write the equation for the plane. Homework Equations The Attempt at a Solution So...
  18. i_hate_math

    Area of Region Vector Calculus

    I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...
  19. toforfiltum

    Approximating equation for tangent plane at a point

    Homework Statement Suppose that you have the following information concerning a differentiable function ##f##: ##f(2,3)=12##, ##\space## ##f(1.98,3)=12.1##, ##\space## ##f(2,3.01)=12.2## a) Give an approximate equation for the plane tangent to the graph of ##f## at ##(2,3,12)##. b) Use the...
  20. toforfiltum

    Confused about partial derivative to function

    Homework Statement Let ##f(x,y) = \|x \| - \|y\| - |x| - |y|## and consider the surface defined by the graph of ##z=f(x,y)##. The partial derivative of ##f## at the origin is: ##f_{x}(0,0) = lim_{h \rightarrow 0} \frac{ f(0 + h, 0) - f(0,0)}{h} = lim_{h \rightarrow 0} \frac {\|h\| -|h|}{h} =...
  21. toforfiltum

    Evaluating limit for this function

    Homework Statement Function is ##lim_{(x,y,z) \rightarrow (0,\sqrt\pi,1)} \ e^{xz} \cos y^2 - x## Homework Equations The Attempt at a Solution As ##x \rightarrow 0## along ##y= \sqrt \pi, z=1##, ##f(x,y,z)= -1## As ##y \rightarrow 0## along ##x=0, z=1##, ##f(x,y,z) = -1## As ##z...
  22. toforfiltum

    Why does the limit not exist?

    Homework Statement This is the function: ##\lim_{(x,y) \rightarrow (0,0)} \frac{(x+y)^2}{x^2+y^2}## Homework Equations The Attempt at a Solution So for ##x \rightarrow 0## along ##y=0##, ##f(x,y)=1## For ##y \rightarrow 0## along ##x=0##, ##f(x,y)=1## also. But the answer says there is no...
  23. toforfiltum

    Does ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## exist?

    Homework Statement Examine the behavior of ##f (x,y)= \frac{x^4y^4}{(x^2 + y^4)^3}## as (x,y) approaches (0,0) along various straight lines. From your observations, what might you conjecture ##\lim_{(x,y) \rightarrow (0,0)} f(x,y)## to be? Next, consider what happens when ##(x,y)## approaches...
  24. toforfiltum

    Equation of ellipsoid and graph

    Homework Statement Equation of ellipsoid is: ##\frac{x^2}{4} + \frac{y^2}{9} + z^2 = 1## First part of the question, they asked to graph the equation. I have a question about this, I know that ##-1\leq z \leq 1##. So what happens when the constant 1 gets smaller after minusing some value of...
  25. toforfiltum

    Describing level surfaces of ##g##

    Homework Statement a) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves only ##x## and ##y## (i.e., ##g (x,y,z)=h (x,y)##). What can you say about the level surfaces of ##g##? b) Suppose ##g## is a function such that the expression for ##g (x,y,z)## involves...
  26. toforfiltum

    Proving a form ##z=f(r)## to be a surface of revolution

    Homework Statement Suppose that a surface has an equation in cylindrical coordinates of the form ##z=f(r)##. Explain why it must be a surface of revolution. Homework Equations The Attempt at a Solution I consider ##z=f(r)## in terms of spherical coordinates. ## p cosφ = f \sqrt{(p...
  27. toforfiltum

    Sketching surfaces described in cylindrical coordinates

    Homework Statement The surface is described by the equation ## (r-2)^2 + z^2 = 1 ## in cylindrical coordinates. Assume ## r ≥ 0 ##. a) Sketch the intersection of this surface with the half plane ## θ= π/2 ## Homework Equations ## r= psin φ ## ## p^2 = r^2 + z^2 ## The Attempt at a Solution...
  28. F

    How do I cross Del with (scalar*vector)?

    Homework Statement Show that for any scalar field α and vector field B: ∇ x (αB) = ∇α x B + α∇ x B Homework Equations (∇ x B)i = εijk vk,j (∇α)i = αi (u x v)i = eijkujvk The Attempt at a Solution Since α is a scalar i wasn't quite sure how to cross it with ∇ So on the left side I have...
  29. M

    I Vector Calculus?

    So I have a quick question that will hopefully yield some clarification. So the divergence of a dyadic ##\bf{AB}## can be written as, $$\nabla \cdot (\textbf{AB}) = (\nabla \cdot \textbf{A}) \textbf{B} + \textbf{A} \cdot (\nabla \textbf{B})$$ where ##\textbf{A} = [a_1, a_2, a_3]## and...
  30. DavideGenoa

    I Magnetic vector potential of infinite straight wire

    The magnetic field generated by an infinitely long straight wire represented by the straight line ##\gamma## having direction ##\mathbf{k}## and passing through the point ##\boldsymbol{x}_0##, carrying a current having intensity ##I##, if am not wrong is, for any point ##\boldsymbol{x}\notin...
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