At a point in an incompressible fluid having spherical polar co-ordinates (r, 0, 4), the velocity components are (2Mr¯3 cos 0, Mr-2 sin 0,0), where M is a constant. a) Show that the velocity is of the potential kind.

A particle is moving along the curve having the vector equation R(t) = 4 cos ti + 4 sin tj + 6tk. Find the magnitude of velocity and acceleration vectors when t = 2

Let C be the curve defined by the following system of equations: √x+6y²-2²=0 3y = z

Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 2x² 5ху + хуz (a) Find the rate of change of the potential at P(5, 4, 7) in the direction of the vector v = i +j - k. (b) In which direction does change most rapidly at P? (c) What is the maximum rate of change at P? Need Help? Watch It Additional Materials lеВook

Sketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =

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?

4. a) Find an equation of the line that is tangent to the curve given by the vector-valued function 7(t) the parametric equations of the line.) b) Find the distance of the point (1,0, – 1) from the plane 2x+y+z+3= 0. t3 t2 3' 2 ? ,t > at the point (,, 1). (You can give =< 3' 2'

3. The position vector of a particle moving in the plane is given by r(t) =e 'i+e'j. a) Find the velocity, acceleration, and speed at an arbitrary time t. b) Sketch the path of the particle together with the velocity and acceleration vectors at time t = 0. Sketch your graph and vectors clearly, labeling all that is relevant.

Determine order and degree * d'y + cos dr? d'y dr? 2 O 5 undefine- Not linear

1)))Assume that the potential on a flat sheet is given by V in volts. a) Determine the maximum rate of change of potential at point (1,2). b) In what direction from the origin does the potential increase and decrease?