State the Fundamental Theorem for Line Integrals

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Todd Williams · Accepted Answer

Step 1To State:Fundamental Theorem for Line IntegralsHere we will use the knowledge of fundamental theorem of calculus in single-variable:∫abg‘(t)=g(b)−g(a)The Fundamental Theorem of Line integrals in some respect is an extension to this theorem in higher dimensionsStep 2The statement of the Fundamental Theorem of Line Integral states that:∫ab△f(r→(t)).r→′(t)=f(r→(b))−f(r→(a))where,1) f is some scalar-valued multivariable function2) ∇f is the gradient of f3) r→(t) is some vector-valued function which parametrizes some path through the input space of fr→(a),r→(b)with end-points4) r→′(t) is the derivative of r→(t)In short the line integral of gradient of a function gives the total change in value of function at the end points.Step 3Answer:∫ab∇f(r→(t)).r→′(t)=f(r→(b))−f(r→(a))

State the Fundamental Theorem for Line Integrals

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