Verifying Green’s TheoremIn Exercises 5–8, verify Green’s Theorem by evaluating both
for the given path.
C: boundary of the region lying between the graphs of
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Calculus
- Green's Second Identity Prove Green's Second Identity for scalar-valued functions u and v defined on a region D: (uv²v – vv²u) dV = || (uvv – vVu) •n dS. (Hint: Reverse the roles of u and v in Green's First Identity.)arrow_forwardThe Fundamental Theorem of Calculusarrow_forwardApplication of Green's theorem Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that JS D Ur Vy dA= u dv, where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forward
- Application of Green's theorem Assume that u and v are continuously differentiable functions. Using Green's theorem, prove that SS'S D Ux Vx |u₁|dA= udv, C Wy Vy where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forwardEvaluate z in rectangular formarrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5 y²dx + 6 x²dy, where C is the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) oriented counterclockwise. f 5 y²dx + 6x²dy =arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ²dx + 2x²dy, where C is the square with vertices (0, 0), (3, 0). (3, 3), and (0, 3) oriented counterclockwise. fy²dx + 2x²dy =arrow_forwardUsing the Fundamental Theorem for line integrals Verifythat the Fundamental Theorem for line integral can be used to evaluatethe given integral, and then evaluate the integral.arrow_forward人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT Find the integral of the vector function F(t)=(f.,cost)arrow_forward
- Answer question in handwritingarrow_forwardEvaluating line integral, can u add explanation to question solutionarrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C6 y2dx+3 x2dy∮C6 y2dx+3 x2dy, where CC is the square with vertices (0,0)(0,0), (3,0)(3,0), (3,3)(3,3), and (0,3)(0,3) oriented counterclockwise.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,