Prove ▽f(r) = (f,(r)/r)r, where r = Vx2 + y2 + ~2 and f'(r)-df/dr is assum...
The answer is neither 1152π nor 1008π (1 point) Vx2 +y2 dA, where D is the domain in Figure 4 I Evaluate F D G:(x- 6)2y2 = 36 Fx2y2 = 144 -R R¢ -R Rf 12 Rg = 6 FIGURE 4 Slp Vx2 +y dA = 1008pi (1 point) Vx2 +y2 dA, where D is the domain in Figure 4 I Evaluate F D G:(x- 6)2y2 = 36 Fx2y2 = 144 -R R¢ -R Rf 12 Rg = 6 FIGURE...
Prove that r'' = r' dr'/dr ? where ' represents derivative wrt time.
12. Let r = Vx2 + y2. Is there an integer n such that r"(-yi + xj) has zero line integral around every closed curve in the Plane?
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F 3. Consider the...
10) Integrate f(x, y) = sin (Vx2 + y2) over the region 0 < x2 + y2 = 16
Question 6 Evaluate SSS, VX2 + y2 +7 av, R is the region between x² + y2 +2=2 and x? ++7=5 (som
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2 4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
Evaluate the line integral ∫ F *dr where C is given by the vector function r(t). F(x, y, z) = (x + y2) i + xz j + (y + z) k, r(t) = t2i + t3j − 2t k, 0 ≤ t ≤ 2
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
Problem 1.20. Let f(z, y)-(X2-y2)/(z2 + y2) 2 for x, y E (0, 1]. Prove that f(x, y) dx dy f f(x,y) dy)dr. Jo Jo JoJo