In this problem you will use the same vector field from Problem 2, namely F(x, y, z) = (3xy, 4xz, -3yz+6) (where you have already verified that div(F) = 0). Do the following: (a) Calculate a vector potential Ā for F. (b) Check your answer by verifying that curl(A) = F. For (a) you can use the step-by-step method from class. Here is a quick review of that method; you can also consult class notes. Consider a C¹ vector field defined for all (x, y, z) = R³, F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) Any vector field A(x, y, z) = (L(x, y, z), M(x, y, z), N(x, y, z)) which is a solution to the vector differential equation curl(A) = F is called a vector potential for F. (We know from class or from Briggs calculus that div(curl(A)) 0, so only a vector field with divergence 0 = can have a vector potential; for this problem, you have already calculated that div(F) = 0 in problem 2). Here is a procedure to compute a vector potential: = Starting from A (L, M, N) where L, M, N are unknown, write the formula for curl(Ã), set it equal to ♬ = (P,Q,R), and separate into three component formulas: one for P, one for Q, and one for R. • Assume L = 0, and use assumption that to simplify the component formulas for P, Q and R. Now you know L. Starting from the formula for Q in step (b), partially integrate with respect to x to get a formula for N, having an integration constant f(y, z) that depends on y and z. Assume f(y, z) = 0. Now you know N. Starting from the formula for R in step (b), partially integrate with respect to x to get a formula for M. You should again have an integration constant g(y, z) that again depends on y and z. This time do NOT assume that g(y, z) is equal to 0. Now you partly know M, except you don't yet have a formula for g(y, z). Starting with the formula for P in step (b), -plug in your formulas for N in step (c) -plug in your formula for M in step (d). - Simplify and solve for g(y, z). (There should be no dependence on x in your formula for g(y, z); if there is, you've done something wrong, so go back and check through your work). Now you know M completely. ⚫ Putting it altogether, you've found a formula for a vector potential A = (L, M, N) of the vector field F.

A particle is moving with velocity V(t) = ( pi cos (pi t), 3t2+ 1) m/s for 0 ≤ t ≤ 10 seconds. Given that the position of the particle at time t = 2s is r(2) = (3, -2), the position vector of the particle at t is?

A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by (x-y,z + y + 3,z²) and the net is decribed by the equation y = 1 - x² - z², y ≥ 0, and oriented in the positive y- = direction. (Use symbolic notation and fractions where needed.) [[ v. ds = Incorrect 8π 3

I am confused on this question. I took the derivative of each section and plugged in 4 and it told me a line cannot be a constant vector. I got L(t)=<10, -1/4, 1>. I am not quit understanding.

Answer both remaining part in 10 minutes in the order to get positive feedback

4. Consider the motion of a particle as described by 1 1 r(t) = = t, Write the acceleration vector of this particle at time t = 1 as a = a+T+ anÑ. You do not need to find T and Ñ, but you can if you find it helpful.

Consider the curve r(t) =(5t² − 3)i + (3t - sin(t))j. - y -5 v(t) = 5 -5 5 10 15 20 (a) Find the velocity of a particle moving along the curve. 25 X (b) Find the acceleration vector of a particle moving along the curve. a(t) =

5. In Newton's theory of gravitation, the Earth's gravitational field is given by, -GM g(F) = p3 where 7 = (x, y, z) is the position vector relative to the centre of the Earth, r = |Fl, M is the mass of the Earth and G is the gravitational constant. The force acting on a particle of mass m at position 7 is given by F = mỹ(r). (a) A rocket of mass m is launched from position (0, 0, R) and travels to (0, 0, R + h) in a time T, along the straight line path C1. A similar rocket travels along the spiral path C2 given by K() = (R+ ) (- 2nt 2nt ř = h(t) : - sin -, 0, cos T T with 0

Match each vector field with its graph. ? 1. F(x, y) = x² + y} ? + 2. H(2,3) = rỉ s ? 3. G(x, y) = y² + x3 A 1111 B (Click on a graph to enlarge it) с

(1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =