Use Stokes' Theorem to evaluate  F · dr  where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices  (3, 0, 0), (0, 3, 0), and (0, 0, 3).

Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.) P(−7, −5, −1), Q(−1, −9, −6) (a) r(t) =  (b) Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)

Let P = (1,3) and Q = (2,6) be points in the plane. Find a vector-valued function R(t) = R0+tv such that R(t) describes the line through P and Q.

4) For the following vector functions, identify the shape of graph they would represent: a. r(t) = (2+ 3t, 5, -2-t) b. r(t) = (2, cos(3t), sin(3t)) c. r(t) = (t,5, t²)