Multivariable Calculus Review Session 4 6. Line Integrals

Multivariable Calculus: Review Session 4 6. Line Integrals Vector Fields 10 A vector field is a function F~ that assigns a vector F~ to each point. F~ (x, y) = hP (x, y), Q(x, y)i 5 F~ (x, y, z) = hP (x, y, z), Q(x, y, z), R(x, y, z)i Conservative / Gradient Vector Fields: F~ = rf for some f . The function f is called a potential for F~ . 0 Ex. F~ (x, y) = h2xy, x2 3y 2 i is a gradient vector field. A potential function is f (x, y) = x2 y y 3 . -5 A gradient vector field F~ = rf is perpendicular to the level sets f (x, y) = k of the potential. -10 -10 -5 0 5 10 Line Integrals of Vector Fields Let ~r(t) = hx(t), y(t)i, a  t  b be a parametrization of a curve C. Line integral of F~ along C: Z Z ~ F · d~r = P dx + Q dy C To compute: d~r = r~0 (t)dt Z C C dx = x0 (t)dt dy = y 0 (t)dt Z b F~ · d~r = F~ (~r(t)) · r~0 (t) dt a Rmk. Similarly for vector fields of three variables. Interpretation: The line integral of F~ along C is the work performed by the force field in moving along C. Ex. 1. Evaluate the line integral R C x2 dx xy dy where C is the arc ~r(t) = h cos(t), sin(t)i, 0  t  ⇡/2. R Ex. 2. Evaluate C y dx + x2 dy where (a) C = C1 is the line segment from ( 4, 7) to (3, 0). (b) C = C2 is the arc of the parabola y = 9 x2 from ( 4, 7) to (3, 0). The Fundamental Theorem for Line Integrals. Let C be a smooth (or piecewise smooth) curve given by the vector function ~r(t), a  t  b. Let f be a di↵erentiable function of two or three variables whose gradient vector rf is continuous on C. Then Z rf · d~r = f (~r(b)) f (~r(a)) C Independence of Path: By the fundamental theorem, if F~ = rf is a gradient vector field, then the line integral depends only on the endpoints of C, not the path itself. The figure at right shows the level sets of a function f (x, y) and its gradient field. C, C1 , and C2 are three paths from the point P to the point Q. Z Z Z rf · d~r = rf · d~r = rf · d~r = f (Q) f (P ) C C1 C2 Ex. 3. Let F~ (x, y) = h3 + 2xy, x2 3y 2 i. R (a) Show that the line integral C F~ · d~r depends only on the endpoints of the path C and not on the path taken between those endpoints. (b) Evaluate Z C F~ · d~r where C is the curve given by ~r(t) = het sin t, et cos ti, 0  t  ⇡. Ex. 4. Evaluate (4, 6, 3). R C F~ · d~r where F~ (x, y, z) = hyz, xz, xy + 2zi and C is the line segment from (1, 0, 2) to Green’s Theorem. Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. Let F~ (x, y) = hP (x, y), Q(x, y)i be a smooth vector field on D. Then ◆ Z ZZ ✓ @Q @P P dx + Q dy = dA @x @y C D Let C be a curve traversed once by ~r(t) = hx(t), y(t)i for a  t  b. C is closed if the initial and terminal points of C coincide, i.e. ~r(a) = ~r(b). C is simple if C has no self-intersections except at the endpoints, i.e. ~r(t1 ) 6= ~r(t2 ) for all a < t1 < t2 < b. C is positively oriented if it is traversed in the counterclockwise direction as t increases from a to b. The region D is always to the left of C. Rmk. The line integral of a vector field along a curve depends on the R orientation of theR curve as follows: If C denotes the curve C traversed in the opposite direction, then C f (x, y) dx = f (x, y) dx. C Z p Ex. 5. Evaluate (3y esin x ) dx + (7x + y 4 + 1) dy where C is the circle x2 + y 2 = 9 traversed in the C counterclockwise direction. R Ex. 6. Evaluate C y 2 dx + 3xy dy where C is the boundary of the region between the circles x2 + y 2 = 1 and x2 + y 2 = 4 in the upper half plane, traversed counterclockwise. Ex. 7. Let C be the triangle with vertices (0, 0), (1, 0), and (0, 1), oriented counterclockwise and let F~ (x, y) = hx3 , xyi. R R (a) According to Green’s Theorem, the line integral C F~ · d~r = C x3 dx + xy dy is equal to a certain double integral. Set up and evaluate this double integral. (b) Verify Green’s Theorem by evaluating the line integral directly. Ex. 8. Evaluate 0  t  ⇡2 . R C F~ · d~r where F~ (x, y) = hxy 2 , x2 yi and C is traced out by ~r(t) = h cos(t), 2 sin(t)i for Ex. 9. Consider the vector field F~ (x, y) = he3x + xy, e3y the unit circle x2 + y 2 = 1 oriented counterclockwise. xyi. Evaluate the integral Z C F~ · d~r where C is R Ex. 10. Evaluate C F~ · d~r where C is the union of line segment C1 from (2, 0, 0) to (3, 4, 5) followed by the vertical line segment C2 from (3, 4, 5) to (3, 4, 0) for the following vector fields F~ . (a) F~ (x, y, z) = hy 2 , 2xy, 4zi (b) F~ (x, y, z) = hy, z, xi