B = < 4xy²z³, 4x²yz, -y²z4-2x²z²> ① check that div(B) = 0 ②Find a vector potential for B.

B = ①check that div (B) = 0 ②Find a vector potential for B.

Suppose that over a certain region of space the electrical potential V is given by the following equation. V(х, у, 2) %3 4x2 - Зху + хуz (a) Find the rate of change of the potential at P(6, 6, 5) in the direction of the vector v = i +j- k. (b) In which direction does V change most rapidly at P? (c) What is the maximum rate of change at P?

Find the scalar equation of the line through (3,-2) and perpendicular to the line r = (4,-3) + t(-5,1) .  Please answer.  Thank you

Transform the equation: 2x2y'' + xy' + 3y = 0 to the form of: a(d2y/dz2) + b(dy/dz) + cy = 0 (a, b, c) = a) (2,1,-3) b) (2,-1,3) c) (-2,1,4) d) (2,-1,4) e) (2,0,0)

find the derivitive, r'(t), of the vector function.  r(t)=a+3tb+t^7c

Line A goes through the points (4, −1, 0) and (1, 5, 2). Line B is parallel to line A and passes through the point (6, 1, −3). Find a parametrization for each of the lines. (Use a different variable for the two parameters.)

22. Given the symmetric equations t = - and t = +', find the vector equation of the line.

Find a vector orthogonal to the line x+y=1

For this problem, enter your vectors with angle-bracket notation: , not in ijk- notation. (a) F = 7xi + (8y – z2) j + (2x + 15z) k - F does not have a vector potential H, H = none (If there is no potential function, enter none for the function.) (b) F = 7xi + (8y – z?) j + (2x – 15z) k F has a vector potential + Ĥ, Ĥ = (If there is no potential function, enter none for the function.)