MATH251 c Justin Cantu Section 16.3: The Fundamental Theorem for Line Integrals Recall: Part 2 of the Fundamental Theorem of Calculus can be written as Z b F 0(x) dx = F (b) − F (a) a where F 0 is continuous on [a; b]. If we think of rf as a sort of derivative of f, we can consider the following as an analogous theorem for line integrals: Fundamental Theorem for Line Integrals: Suppose C is a smooth curve defined by r(t), a ≤ t ≤ b. Let f be a function whose gradient vector field rf is continuous on C, then Z rf · dr = f(r(b)) − f(r(a)) C This theorem says that for gradient vector fields, the line integral only depends on the endpoints of Z the curve C, and not on the path itself. We say rf · dr is independent of path. C Z Example 1: Let f(x; y) = 3x − xy + x2y. Evaluate F · dr, where F = rf and C is a curve given C by r(t), 1 ≤ t ≤ 2, where r(1) = h2; 1i and r(2) = h4; 4i. 1 MATH251 c Justin Cantu Note: Not all line integrals are independent of path. Z Example 2: Evaluate F · dr, where F(x; y) = hy2; xi and C (a) C is the line segment from (0; 0) to (2; 4). (b) C is the arc of the curve y = x2 from (0; 0) to (2; 4). It can be shown line integrals of gradient vector fields are the only ones independent of path. Be- cause of the Fundamental Theorem for Line Integrals, it will be useful to determine whether a given vector field F corresponds to a gradient vector field. A vector field F is called a conservative vector field if it is the gradient of some scalar function f, that is, if there exists a function f such that F = rf In this situation f is called a potential function for F. Example 3: Show that F(x; y) = hy; xi is conservative. 2 MATH251 c Justin Cantu Conservative Test (R2): Let F = hP; Qi be a vector field on an open simply-connected region D, where P and Q have continuous first partial derivatives. Then @P @Q = @y @x if and only if F is conservative. Intuitively, a region is open if it doesn't include any of its boundary points, and simply-connected if it is a single \piece" with no holes. Typically, D will be all of R2, which is an open simply-connected region. Example 4: Is F(x; y) = h3x2 − 2y2; 4y2 − 2xi a conservative vector field? If so, find a potential function for F. Example 5: Is F(x; y) = hx + y; x − 2i a conservative vector field? If so, find a potential function for F. 3 MATH251 c Justin Cantu Z We now have two methods to calculate F · dr: C 1. If F is conservative, we can find a potential function f (so that F = rf) and apply the Fundamental Theorem for Line Integrals: Z Z F · dr = rf · dr = f(r(b)) − f(r(a)) C C 2. We can substitute in terms of t: Z Z b F · dr = F(r(t)) · r0(t) dt C a Z Example 6: Evaluate F · dr, where F(x; y) = h2xy3; 3x2y2i and C is the curve defined by C r(t) = ht3 + 2t2 − 4t; t4 − 3t2i, 0 ≤ t ≤ 2. 4 MATH251 c Justin Cantu Z 1 Example 7: Evaluate F · dr, where F(x; y) = h3 + 2xy2; 2x2yi and C is the arc of y = from C x 1 (1; 1) to 4; . 4 Z Example 8: Evaluate F · dr, where F(x; y) = h3x2 − 4y; 4y2 − 2xi and C is the curve given by C r(t) = ht3 + 1; t2i, 0 ≤ t ≤ 1. 5 MATH251 c Justin Cantu The Fundamental Theorem for Line Integrals also applies to vector fields over R3. We'll see a conservative test for these vector fields in Section 16.5. Z Example 9: Given that F(x; y; z) = h4xez; cos(y); 2x2ezi is conservative, evaluate F · dr, where π C C is the curve given by r(t) = hsin(t); t; cos(t)i, 0 ≤ t ≤ . 2 6.