Evaluate the Laplacian of the function zx2 x2 + y2+ z2 p(x, y,z)- (a) directly in Cartesian coordinates, and (b) after changing to a spherical polar coordinate system. Verify that, as they must, the two methods give the same result.

The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))j Find the velocity vector of the point. v(t) = Find the acceleration vector of the point. a(t) = Find the speed of the point. s(t) = =

Parametrize the intersection of the surfaces y² - z² = x - 6, y² + z² = 25 using trigonometric functions. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the parametrization of the y variable in the form a cos (t).) x(t) = y(t) = z(t) =

Parametrize the intersection of the surfaces y- z = x - 5, y + z? = 4 using trigonometric functions. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the parametrization of the y variable in the form a cos (1).) x(t) = y(t) = z(t) =