Recent content by sunnyday11

Homework Statement F(x,y,z) = zy i + z k and the surface S defined by x2 + y2 + z2 = 4, z\geq\sqrt{3} Homework Equations The Attempt at a Solution Using line integral method, I got -\sqrt{3}\pi But using the flux of curl method, I got curl = y j - z k Then I change...

Homework Statement Let T: U-->V be a linear map between vector spaces U and V and let E be basis for U and F be a basis for V. Explain what is meant by the matrix m(T)^{F}_{E} of T taken with respect to E on the left and F on the right. Homework Equations The Attempt at a Solution...

x3 + x - a = 0 Sorry, I am not quite familiar with the complex plane, can you give me a hint of theorem you mentioned?

The roots are 0 and i. So for injectivity, h is not injective since both 0 and i lead to the same result; f on the other hand is injective since only 0 is in its field. For surjectivity, I still can't think of anything to prove.

Homework Statement Decide with proof whether the mapping is injective and/or surjective. Let f: A-->B be a mapping. h: C--> C; h(x)=x3 + x (complex field) f: Z--> Z; h(x)=x3 + x (integer field) Homework Equations injective means f(a)=f(a') => a=a' surjective means...

Homework Statement F(x,y) = y i + (x2y + exp(y2)) j Curve C begin at point (0,0) go to point (pi, 0) along the straight line then go back to (0,0) along curve y=sin(x) Find circulation of F around C Homework Equations The Attempt at a Solution Curve part 1 Using Green theorem I got...

Thank you! Yes I forgot to consider the normal after computing the curl. So I get to surface integral of 1 which translates into double integral of 1 over the region enclosed by S and since x and y form a circle the region is the area of a circle which is 9 pi.

Homework Statement F = ( 2y i + 3x J + z2 k where S is the upper half of the sphere x2 + y2 + z2 = 9 and C is its boundary. Homework Equations The Attempt at a Solution I used Stoke's Theorem and found the solution to be 36 pi, but when I use line integral to verify, using...

Homework Statement F(x,y,z) = (2x-z) i + x2y j + xz2 k and the volume is defined by [0,0,0] and [1,1,1]. Homework Equations flux integral = \int\int\int div F dV The Attempt at a Solution \int\int\int div F dV = \int\int\int (2+x2-2xz)dxdydz = 2 + 1/3 - 1/2 = 11/12 But I...

Thank you very much for your help! I got 1/15 for both parts, so I just subtracted one from the other and got 0 which agrees with the other solution.

Homework Statement Verify Green's Theorem for F(x,y) = (2xy-x2) i + (x + y2) j and the region R which is bounded by the curves y = x2 and y2 = x Homework Equations \int CF dr = \int\intR (dF2/dx - dF1/dy) dxdy The Attempt at a Solution For \int CF dr , r(t) = x i + x2 j...

Homework Statement Verify Stoke's Theorem for F = y i + z j + x k and S the paraboloid z=1-(x^2+y^2) with z bigger or equal to 0 oriented upward, and the curve C which is the boundary of S. Homework Equations Stoke's Theorem line interal = flux of curl of F (don't know how to...

By induction k-1 is odd. So, gap(i,j)=(i,i+1) (odd number of adjacent transpositions) (i,i+1) = 1+ odd# + 1 = odd # Thank you so much, but do you know how can I put this in a more formal presentation?

Homework Statement Prove that any transposition is a product of an odd number of adjacent transposition. Homework Equations The Attempt at a Solution Let x=(i,j) Define gap(x) = j-i By induction on gap: If gap(x)=1 then already adjacent. Suppose k= j-i...