Chapter 16 – Vector Calculus

Chapter 16 – Vector Calculus 16.3 The Fundamental Theorem for Line Integrals • Objectives: • Understand The Fundamental Theorem for line integrals • Determine conservative vector fields 16.3 The Fundamental Theorem for Line Integrals

FTC – Part 2 • Recall from Section 5.3 that Part 2 of the Fundamental Theorem of Calculus (FTC2) can be written as: where F′is continuous on [a, b]. 16.3 The Fundamental Theorem for Line Integrals

Fundamental Theorem for Line Integrals • Let C be a smooth curve given by the vector function r(t), a ≤ t ≤ b. • Let f be a differentiable function of two or three variables whose gradient vector is continuous on C. • Then, 16.3 The Fundamental Theorem for Line Integrals

Note • Theorem 2 says that we can evaluate the line integral of a conservative vector field (the gradient vector field of the potential function f) simply by knowing the value of f at the endpoints of C. • In fact, it says that the line integral of f is the net change in f. 16.3 The Fundamental Theorem for Line Integrals

Note: • If f is a function of two variables and C is a plane curve with initial point A(x1, y1) and terminal point B(x2, y2), Theorem 2 becomes: 16.3 The Fundamental Theorem for Line Integrals

Note: • If f is a function of three variables and C is a space curve joining the point A(x1, y1, z1) to the point B(x2, y2, z2), we have: 16.3 The Fundamental Theorem for Line Integrals

Paths • Suppose C1 and C2 are two piecewise-smooth curves (which are called paths) that have the same initial point A and terminal point B. • We know from Example 4 in Section 16.2 that, in general, 16.3 The Fundamental Theorem for Line Integrals

Conservative Vector Field • However, one implication of Theorem 2 is that whenever fis continuous. • That is, the line integral of a conservativevector field depends only on the initial point and terminal point of a curve. 16.3 The Fundamental Theorem for Line Integrals

Independence of a Path • In general, if F is a continuous vector field with domain D, we say that the line integral is independent of pathif for any two paths C1 and C2 in D that have the same initial and terminal points. • This means that line integrals of conservative vector fields are independent of path. 16.3 The Fundamental Theorem for Line Integrals

Closed Curve • A curve is called closedif its terminal point coincides with its initial point, that is, r(b) = r(a) 16.3 The Fundamental Theorem for Line Integrals

Theorem 3 • is independent of path in Dif and only if: for every closed path C in D. 16.3 The Fundamental Theorem for Line Integrals

Physical Interpretation • The physical interpretation is that: • The work done by a conservative force field (such as the gravitational or electric field in Section 16.1) as it moves an object around a closed path is 0. 16.3 The Fundamental Theorem for Line Integrals

Theorem 4 • Suppose F is a vector field that is continuous on an open, connected region D. • If is independent of path in D, then F is a conservative vector field on D. • That is,there exists a function f such that f=F. 16.3 The Fundamental Theorem for Line Integrals

Determining Conservative Vector Fields • The question remains: • How is it possible to determine whether or not a vector field is conservative? 16.3 The Fundamental Theorem for Line Integrals

Theorem 5 • If F(x, y) = P(x, y) i + Q(x, y) jis a conservative vector field, where P and Q have continuous first-order partial derivatives on a domain D, then, throughout D, we have: 16.3 The Fundamental Theorem for Line Integrals

Conservative Vector Fields • The converse of Theorem 5 is true only for a special type of region, specifically simply-connected region. 16.3 The Fundamental Theorem for Line Integrals

Simply Connected Region • A simply-connected regionin the plane is a connected region Dsuch that every simple closed curve in D encloses only points in D. Intuitively, it contains no hole and can’t consist of two separate pieces. 16.3 The Fundamental Theorem for Line Integrals

Theorem 6 • Let F = Pi + Qjbe a vector field on an open simply-connected region D. Suppose that P and Q have continuous first-order derivatives and throughout D. • Then, F is conservative. 16.3 The Fundamental Theorem for Line Integrals

Example 1 – pg. 1106 • Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 16.3 The Fundamental Theorem for Line Integrals

Example 2 • Find a function f such that F=f. • Use part (a) to evaluate along the given curve C. 16.3 The Fundamental Theorem for Line Integrals

Example 3 – pg. 1107 # 24 • Find the work done by the force field F in moving an object from P to Q. 16.3 The Fundamental Theorem for Line Integrals

Example 4 • Determine whether or not the given set is • open, • connected, • and simply-connected. 16.3 The Fundamental Theorem for Line Integrals

Example 5 – pg. 1106 # 12 • Find a function f such that F=f. • Use part (a) to evaluate along the given curve C. 16.3 The Fundamental Theorem for Line Integrals

Chapter 16 – Vector Calculus