Problem 1.32 Check the fundamental theorem for gradients, using T = x² + 4NN +2yz'the points a = D, and the three paths in Fig. 1.28: (0.0.0), b= ( Ka) (0.0,0)→IKI,0.0) → (1,.0)→ (1,1.1). (b) (0,0, 0) → KOJ O, 1) → (0.,1.)-110E (c) the parabolic path z = xAN=X (a) (b) (c)

B=re-²9, find curl(B) analytically. Compare the value of the divergence at points P1 = (x1, y1) = (0.5, 0.5) and P2 = (x2, y2) = (3, 2).

Please show complete answer

Evaluate the given integral. S 1 (x²+4)3/2

1.1 Let's say a cart's position is given by x(t) at3 – Bt. (1) - Assume the constants, a and B, are positive. (a) What must be the general type of unit (like length, mass, time, and combinations of these) for each of the constants, a and B? (b) Draw, by hand, a sketch of the two functions at and –ßt for t > 0. Add the two curves graphically to get a final sketch of x(t). (c) Calculate the cart's velocity vr and acceleration ar. (d) Find an expression for the time t = to at which the cart is momentarily at rest. Check: If a and B had the values 1 and 27, respectively (in the correct SI units), you would find t = 3 s. (e) Give the expression for the cart's position x(to) at the time t = to. Always simplify your answer, - and confirm that the units are correct.