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Unformatted text preview: LESSON 20 Fundamental Theorem for Line Integrals READ: Section 17.3. NOTES: One version of the Fundamental Theorem of Calculus says Z b a f ( x ) dx = f ( b )- f ( a ) , provided y = f ( x ) is continuous on the interval of integration. This formula shows there is a connection between the operations of integration and differentiation. It also provides a practical method of evaluating certain integrals (namely integrals of functions which happen to be derivatives of other functions). There is a similar formula involving line integrals. The Fundamental Theorem for Line Integrals concerns the line integral of the gradient of a function of several variables, such as w = f ( x,y,z ). Suppose f is differentiable in a region that includes a smooth curve C parameterized by a function-→ r ( t ), for a ≤ t ≤ b . Suppose that the gradient of f , ∇ f (-→ x ), is continuous on C . (These requirements will be met by almost any function f we are likely to bump into in the text.) Then Z C ∇ f · d-→ r = f (-→ r ( b ))- f (-→ r ( a )) . Note that-→ r ( b ) and-→ r ( a ) represent to two ends of the curve C , and so f (-→ r ( b ))- f (-→ r ( a )) is the difference of the values of f and the terminal point of the curve and the initial point of the curve. Doesn’t that remind you the formula at the top of the page? There we evaluated the integral by taking the difference of the function values at the terminal point and the initial point of the interval. The easy proof is given starting on page 944 of the text. The formula above works only for line integrals of gradient vector fields! There are a few consequences of the formula that you ought to be aware of. First, note that the right hand side of the formula depends only on the end points of the curve C . So if we know that the curve C starts at the point...
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