2. Problems on Gradient and Divergence Operators Suppose that r is the position vector; that is, r = re, + ye, + ze;, and that r is its magnitude, i.e., r² = x² + y? + 22. Prove the following identities: V.r = 3, V xr = 0, v () --; r p3

1. A scalar function, V is given by V = x y zº. Find the gradient of V, that is, VV. 2. If a vector, B = 4xy³ā, +2y³ā, +x²yza,, find the divergence of B, that is, V.B.

Consider the function f(x,y) = x² - 4y² - 9 and the point (-1,1). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. a. What is the unit vector in the direction of steepest ascent at P? (Type exact answers, using radicals as needed.) What is the unit vector in the direction of steepest descent at P? (Type exact answers, using radicals as needed.) b. Which of the following vectors is in a direction of no change of the function at P? O A. (2,-8) OB. (8,2) OC. (-2,8) D. (8,-2)