I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
Question about conditions for conservative field
In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open.
Usually in textbooks there is not much explanations on why these...
Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)?
Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...
Homework Statement
For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get
∇ x (∇ x E) = ∇ x -∂B/∂t
I feel like it'd be very wrong to pull out the time derivative. Am I correct?
If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0
I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
Homework Statement
Given an electric field in a vacuum:
E(t,r) = (E0/c) (0 , 0 , y/t2)
use Maxwell's equations to determine B(t,r) which satisfies the boundary condition B -> 0 as t -> ∞
Homework Equations
The problem is in a vacuum so in the conventional notation J = 0 and ρ = 0 (current...
Dear Physics Forum personnel,
I am a college sophomore with double majors in mathematics & microbiology and an aspiring analytic number theorist. I will be going to self-study the vector calculus by using Hubbard/Hubbard as a main text and Serge Lang as a supplement to Hubbard; this will help...
Dear All,
I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
Homework Statement
Let ## E ## be the ellipsoid:
$$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$
Let ## S ## be the part of the surface of ## E ## defined by:
$$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$
Let F be the vector field defined by $$ F=(-y,x,0)$$
A) Explain why ##...
Homework Statement
Given the eqn x=2, y=sin(t), z=cos(t), draw this function in 3-space.
Homework Equations
ABOVE^
The Attempt at a Solution
I did this:
x^2+y^2+z^2=2^2+(sin(t))^2+(cos(t))^2=5
Therefore we get x^2+y^2+z^2=5
Which is the eqn of a sphere with radius root5.
My friend said it's...
Let f: \mathbb R^2 \to \mathbb R^2 given by f=(sin(x-y),cos(x+y)) : find the equation of the tangent plane to the graph of the function in \mathbb R^4 at (\frac{\pi}{4}, \frac{\pi}{4}, 0 ,0 ) and then find a parametric representation of the equation of the tangent plane
What I did: the...
Hi, I want to translate this equation
R_{\hat{n}}(\alpha)\vec{x}=\hat{n}(\hat{n}\cdot\vec{x})+\cos\left(\alpha\right)(\hat{n}\times\vec{x})\times\hat{n}+\sin\left(\alpha\right)(\hat{n}\times\vec{x})
to index notation (forget about covariant and contravariant indices).
My attempt...
Homework Statement
By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that
∇ × (φ∇φ) = 0,
where φ(r) is any scalar field.
Homework Equations
∇×(φA) = (∇φ)×A+φ(∇×A)?
The Attempt at a Solution
I honestly have no idea how to even...
Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables.
My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...