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!
Homework Statement
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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
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All the identities given at...
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...
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...
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...
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...
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...
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...
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...
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...
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
$$V(r,h)=\pi r^2h$$
$$2...
Homework Statement
Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point ##(x,y)## is given by ##M(x,y)=3x^2+y^2+5000##. If Igor is at the point ##(8,6)##, describe the curve along which he should travel if he wishes to reduce the field...
Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x axis and similarly x-z plane rotating wrt to y axis.
My question is...
Homework Statement
Let ##F(x,y)=4sin(xy)+x^3+y^3## Use Newton's method to approximate the critical point that lies near ##(x,y)=(-1,-1)##
Homework Equations
The Attempt at a Solution
I have a problem here because the derivative is not a square matrix. Hence, I can't find the inverse needed...
Homework Statement
Let $$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...
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...
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...
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...
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...
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} =...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...