What does a "nonsingular derivative" mean. It comes in the following context: "If f: R^2 --> R^2 is a function with a nonsingular derivative everywhere, is f bijective?"
Here is a start of my solution which I am stuck with.
Consider this continuous surface f(x,y) = z. Consider arbitrary surface point (i,j,z _i_j). This point must exist (given f(i,j) = z _i_j).
Proof by contradiction:
Consider continuous functions f(x,j) and f(i,y) which are curves that...
Can you guys help me prove: Given a continuous and differentiable function (or surface) f: R^2 -> R, such that f(x,y) = z ... contour lines can always be drawn... the function is NOT bijective.
I've been thinking of choosing any arbitrary point and showing that the curves that intersect to...
Why do you guys think that given two 3x3 matrices, they are similar if and only if their characteristic polynomial and minimal polynomial are equal (this reasonably fails for 4v4 matrices though)?
Homework Statement
a. An infinite cyclindrically symmetric current distribution has the form
\vec J (r, \phi, z) = J_0 r^2/R^2 \ \ \ \vec\hat \phi for R<r<2R. Outside the interval, the current is 0. What is the field everywhere in space?
b. An infinite cyclindrically symmetric current...
Ok here is one:
If there exists linear functional g on V such that g(\{ w_1, w_2, w_3,..., w_n, v_n_+_1, v_n_+_2, ... , v_r \}) = k_1*w_1 + k_2*w_2 + .... + k_r*v_r where r = dim V and for i=1 to r, k_i, w_i, v_i \in K field
This g is I think called a linear form and when it is...
That is of course the basis extension principle (theorem?).
So here is my revised take on the Ans (please tell me if I am reasoning it):
Take some B_W = { w_1, w_2, w_3, ..., w_n} , n = dim W <= dim V
B_W \subset B_V for some Basis of V.
We know f(p*w1 + q*w2) = p*f(w1) + q*f(w2)...
I am sorry, I misworded my question. At any rate, what I wanted to know was a confirmation of basis extension. For W < V, B_W (some basis of W) is linearly independent and spans W, therefore, there exists some B_V (basis of V) s.t. B_W < B_V.
I believe this is the basis extension theorem...
I am writing a solution for the following problem, I hope someone can correct it, because I am not sure what I am missing.
Q. V is a finite dim. vsp over K, and W is a subspace of V. Let f be a linear functional on W. Show that there exists a linear functional g on V s. t. g(w)=f(w).
Ans...
This is a problem was thinking about.
If I have a capacitor and fill it with a dialectric of some dialectric constant (the dialectric fits perfectly and is mounted via a frictionless bearing so it can move freely). How fast does it move?
This should be easy, but for some odd reason I am not getting the right answer.
Assuming the potential V=0 at infinity, what is the V at the center of a sphere with volume charge density rho(r) = rho_0 * R/r
I keep getting (integral from 0 to R K*(4*pi*rho_0)*R/2) which I dont think is...