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1. Nonsingular derivative

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?"
2. That given a continuous surface, contour lines exist

The function is everywhere differentiable and continuous/
3. That given a continuous surface, contour lines exist

Hmm.... I can't just use the curves along x and y axes. Because there are bound to be such R^2 -> R functions which elude only those particular axes?
4. That given a continuous surface, contour lines exist

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...
5. That given a continuous surface, contour lines exist

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...
6. 3x3 similar matrices defined by characteristic and minimal polynomials

Not exactly sure what you mean. How do Jordan blocks get involved?
7. 3x3 similar matrices defined by characteristic and minimal polynomials

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)?
8. Cylindrically symmetric current distribution: Magnetic field in all space

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...
9. Basis of a subspace of a vectorspace

Sorry, I meant to say: for some B_W \subset B_V for some basis of V.
10. Linear functional equivalence in vsp and subsp

Is it possible to just define g(w') to be 0? It seems too simple.
11. Linear functional equivalence in vsp and subsp

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...
12. Linear functional equivalence in vsp and subsp

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)...
13. Basis of a subspace of a vectorspace

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...
14. Linear functional equivalence in vsp and subsp

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...
15. Basis of a subspace of a vectorspace

Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?
16. Interesting problem - movement of dialectric

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?
17. NONUNIFORM Vol. Charge Density - V at the center of sphere

I did the problem again, now I have: V_0 = k*Q/R = k*4*pi*rho_0*R^2/2 = rho_0*R^2/(2*epsilon_0) Can anyone confirm if I did this right?
18. NONUNIFORM Vol. Charge Density - V at the center of sphere

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...