(a) You should already be able to prove, by direct computation of the required differ- entiations in Cartesian coordinates, that the curl of a gradient is always zero and the divergence of a curl is always zero. In a similar manner, verify the following identities in which and are arbitrary differentiable scalar functions of position, and F and G are arbitrary differentiable vector functions of position: (i) ▼ · (F) = &V · F + F · V (ii) ▼ × (▼ × F) = V (VF) – V²F - (b) Calculate the divergence of the function: F(x, y, z) = f(x)â + ƒ(y)ŷ + ƒ(−22) and show that it is zero at the point (c, c, -c/2). [Hint: chain-rule.] (c) Calculate the curl of the following function. -yzx+xzy in Cartesian coordinates. Now write the same equation in cylindrical coordinates and calculate its curl in cylindrical coordinates. Convert your answer back to Cartesian coordinates and compare to the answer you obtained previously. (Note: all information required can be found on the inside covers of Griffith's.)

Suppose that f(z,y) is differentiable, and suppose that the directional derivative of f at the origin attains a maximum value of 5 in the direction of the vector from the origin to the point (-3,4). Find Vƒ(0,0).

Consider the function defined by f(x,y) = 4xy - 2x2. Evaluate the directional derivative of f at point(1,-1) in direction of the vector which subtends an angle of pi/6 with the positive x-axis.

Let h(x, y) be a differentiable function and let (xo, yo) be a point in the domain of h. Also, let u be a unit vector. Then D- h(xo, yo) = -Du h(xo, yo). Hint: Your job is to show that two directional derivatives are related. First notice the two negative signs. One of them is the negative of a VECTOR. Start by computing the directional derivative on the LHS. Remember that for the dot product, a (-b) = (-a) b=-(ab).

a. Sketch the graph of r(t) = ti+t2j. Show that r(t) is a smooth vector-valued function but the change of parameter t = 73 produces a vector-valued function that is not smooth, yet has the same graph as r(t). b. Examine how the two vector-valued functions are traced, and see if you can explain what causes the problem.

(a) Find the directional derivative of z = x²y at (5,5) in the direction of π/2 with the positive x-axis. (b) In which direction is the directional derivative the largest at the point (5, 5)? Enter your answer as a vector whose length is the largest value of the directional derivative.

Use the gradient vector to find the equation of the tangent plane to the surface x? + y? - at the point (2,2,4). Write your answer in the form Ax + By + Cz = D.

The 1-variable chain rule (stated in class but not yet proved): Let f(x1,...,xn) be a differentiable function defined on a subset DCR", with gradient vector Vƒ (p) at each pЄ D. Let y(t) = (v1(t), ..., Yn(t)) be a differentiable path defined on some interval a < t < b, with derivative vector 7′(t) = (v(t), ..., n(t)) for each t. Suppose that y(t) = D for each t. Prove that Exercises: d fox(t) =Vf(())-7 (0) dt о (for a

Select ALL true statements from the following list. (I) The directional derivative Duf(a, b) is parallel to the vector u. (II) If u is perpendicular to the gradient vector Vƒ(a, b), then Duf(a, b) = 0. (III) If f(x, y) is differentiable at (a,b) and has fx(a,b) = 0 and fy (a, b) = 0, then z = f(x, y) has a horizontal tangent plane at (a, b). A. (II) only B. (I) and (II) only C. (I) and (III) only D. (II) and (III) only E. (I), (II), and (III)

Compute the derivative of the vector-valued function r(t)=⟨sin2t,et2⟩.