Section 16 3 The fundamental theorem for line integrals MATH 251 Fall 2023 De nition Let C be a 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 ZC rf dr f r b f r a De nition A vector eld F is said to be conservative if F is the gradient vector eld of some scalar function f that is F rf In this case f is called a potential function for F If F is a conservative vector eld with a potential function f then the line integral of F along a curve C given by the vector function r t a t b is ZC rf dr ZC F dr line integral z f r b f r a di erence of f at endpoints z Remark only to know its potential function and the endpoints of the curve If F is a conservative vector eld then to evaluate the line integral of F over a curve we need If two smooth curves C1 and C2 have the same start point and the same endpoint then the line integral of the conservative vector eld F is the same over each curve This is called path independence Example 1 Find the work done by the vector eld in moving a particle from the point 2 4 5 to the point 1 0 p3 F x y z 5 x2 y2 z2 3 2hx y zi 1 2 Independence of path Suppose C1 and C2 are piecewise smooth curves also called paths that have F dr For example take the same initial A and terminal point B F x y hy2 xi and consider C1 the line segment from 4 0 to 0 2 and C2 the arc of the parabola x 4 y2 from 4 0 to 0 2 3 while Indeed you can check for yourself that RC1 RC2 De nition We say that the line integralRC F dr is independent of path ifRC1 any two paths C1 and C2 sharing the same initial and terminal points In general RC1 F dr 6 RC2 F dr RC2 F dr 4 F dr 8 3 F dr for As we have seen above line integrals of conservative vector elds F are independent of path because to compute the integral of F over a curve C we only need to plug the endpoints of C into a potential function f of F and nd the di erence Consequently we do not need to parameterize the curve C De nition A curve is called closed if its terminal point coincides with its initial point i e r a r b Observation If F is conservative and C is a closed curve thenRC F dr 0 Goal As conservative vector elds are convenient our next task is to determine whether a vector eld F is conservative Theorem conservative test Suppose that F x y P x y i Q x y j and P Q have continuous rst order partial derivatives on an open simply connected domain D Then F is conservative if and only if P y Q x Example 2 Determine whether or not F x y hx2 2y 3x 2xyi is conservative Example 3 Determine whether F x y hx 2y 2x y2i is conservative 3 Here are the steps to nd a potential function f for F x y P x y i Q x y j when we know that F is conservative The steps are for two dimensional vector elds 1 Step 1 Write fx P x y and fy Q x y h y where h y is a function of y to be found in the next step 2 Step 2 Integrate both sides of fx P x y with respect to x to have f x y R P x y dx 3 Step 3 Di erentiate both side of f x y R P x y dx h y with respect to y and use the equation fy Q x y to nd h y 4 Step 4 Conclude F x y Example 4 Find a potential function f for the vector eld F from Example 3 4 Example 5 Let F x y h3x2y4 4x3y3i Evaluate RC F dr where C is the curve given by r t ht3 4t2 3t2 2ti for 0 t 2 Example 6 Let F x y h2xy 2y3 x2 6xy2i EvaluateRC F dr where C is the arc y 2x from the point 0 1 to 1 2 Example 7 Let F x y h2x 4y 2x2i EvaluateRC F dr where C is the line segment from 0 1 to 2 5 5 Example 8 Given that F x y z h2xy exz x2 exi is conservative nd its potential function Then evaluateRC F dr where C is the curve r t t sin t 2 1 for 0 t 1
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