Reference no: EM13373810
(1) Prove the following vector calculus identity in R3, where f is a twice continuously dierentiable scalar eld and F is a twice continuously dierentiable vector fi eld:
![672_Find the eigenvalues and eigenvectors4.png](https://secure.expertsmind.com/CMSImages/672_Find%20the%20eigenvalues%20and%20eigenvectors4.png)
(2) Let F(x; y; z) = (y + z)i + (x + z)j + (x + z)k. The sphere x2 + y2 + z2 = a2 intersects the postive x-, y-, and z-axes at points A, B, and C, respectively. The simple closed curve K consists of the three circular arcs AB, BC, and CA. Let S denote the surface ABC of the octant of the sphere bounded by K, oriented away from the origin. Let T denote the unit tangent vector to K, and n the unit normal vector to S.
(a) Calculate the line integral
ds directly without using Stokes' Theorem.
(b) Calculate the surface integral
without using Stokes' Theorem.
(3) Consider the matrix ![1329_Find the eigenvalues and eigenvectors2.png](https://secure.expertsmind.com/CMSImages/1329_Find%20the%20eigenvalues%20and%20eigenvectors2.png)
(a) Find a PLU decomposition for the matrix Z.
(b) Use your answer to part (a) to solve the system of equations
x + 3y + z = 1;
2x + 6y + 4z = 4;
x + 2y + 2z = 6:
(4) Consider the matrix ![1127_Find the eigenvalues and eigenvectors3.png](https://secure.expertsmind.com/CMSImages/1127_Find%20the%20eigenvalues%20and%20eigenvectors3.png)
(a) Find the eigenvalues and eigenvectors of A(t).
(b) For which values of t will A(t) be diagonalisable? Explain.
(c) For those values of t from part (b), nd an invertible matrix P(t) and a diagonal matrix D(t) such that A(t) = P(t)D(t)P(t)-1.
(d) Give a formula for A(t)n (where n is a positive integer) that holds for those values of t from part (b).