Problem 1 (a). Make a sketch of the 1-form w = (l sin(x)| + 1) dx supported on U = [-r/2.2 [2 marks]
1/2 sin sin-1 x (d) L dx V1 -2
Problem 6. Describe the surface r(u, u)-R cos u x + R sin u ý + uz where 0 < u < 2π and 0 < u-H. and R and H are positive constants. What is the surface element and what is the total surface area? Show that Or/au, or/àv are continuous across the "cut at 2T coS W T
1. a) Substitute u = sin(x) to evaluate sin^2(x) cos^3(x) dx. [trig identity sin2(x)+cos2(x) = 1]. b) Find the antiderivatives: i) sin(2x) dx ii) (cos(4x)+3x^2) dx
Am = } $(w). cos(mkr)dx Bm= f(x) = sin(mkr)dx - Given the periodic quadratic periodic function f(x) = G) "for - <x< . Calculate Ag. There is a figure below that you should be able to see. You may (may not) need: Jup.sin(u)du = (2-u?)cos(u) +2usin(u) /v2.cos(u)du = 2ucos(u)+(u2–2)sin(u) -N2 0
Consider the following initial value problem: dy = sin(x - y) dx, y(0) 1. Write the equation in the form ay = G(ax +by+c), dx where a, b, and c are constants and G is a function. 2. Use the substitution u = ax + by + c to transfer the equation into the variables u and x only. 3. Solve the equation in (2). 4. Re-substitute u = ax + by + c to write your solution in terms...
Problem 6 Evaluate: dx 16 - 22 Hints: sin²x = 1 - cos(2.0) and sin(2x) = 2 sin c cos r. 2 (Show all details.)
Problem 1. Show that the eigenvalue problem -X"(r) - XX(X), X(-) = X(L),X'(-) = X(L) has the following eigenvalues and eigenfunctions An - (92), X,(w) -- sin (7+), xy(x) = cos ("E") - - 0,1,2,...
= r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2 = r.Cos (0), y r sin(0), and zr0 Let x.y,z)=x y+y zxz, where x 3-where w(r,0) = u(x(r,0),y(r,0),2(r,0)) Owr.0) for r= 1, 0 = д0 (r,0) and дr Evaluate 2
Consider the following boundary value problem: du du dx dx u=-e* sin(x) Discretize the ODE using backward second-order accurate scheme for both derivatives. The second order finite accuracy difference for the derivatives are given by: 2h (3)-1(1,2)-45 (7.1)+31(x) 8 (*)== (4.5) +41 (1.2) -51 (3.1) +2f (x) h?
6.8. Verify that u(x, y)= A sin(27Tx) sin(27ty) solves Poisson's equation V2u-Ron W (0, 1) x (0,1) for some A-value, where R(x, y) sin(2x) sin(2Ty) (a) Find the correct A value (b) Compute the total source S w RdA (c) Compute the flux out through the top part of W (y 1) and verify by symmetry that it is one-quarter that of the full source S. 6.8. Verify that u(x, y)= A sin(27Tx) sin(27ty) solves Poisson's equation V2u-Ron W (0,...