Find the gradient and Laplacian of the scalar field f in two dimensions, given in the Cartesian coordinates as: f(x, y) = y sin(x), and show that the Laplacian of f is equal to - f. Find the directional derivative of f at the point P(π/6,5) in the direction of a = 3i+4j.

Let 7 (t) = (3t3 + 4, – 4ešt, – 5 sin(4t)) Find the unit tangent vector T (t) at the point t = 0 T(0) = <

Let r(t)=sin(t + 1) + sin(t + 1)ĵ + √2 cos(t + 1) * denote the position of an object. (a) Find a formula for the unit tangent vector T(t). (b) Find the tangential component a(t) of the acceleration.