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Verifying Stokes’s Theorem In Exercises 3-6, verify Stokes’s Theorem by evaluating
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Chapter 15 Solutions
Multivariable Calculus
- Use Stokes' Theorem to evaluate F• dr where C is oriented counterclockwise as viewed from above. (x + y?)i + (y + z?)j + (z + x2)k, C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3). F(x, у, z)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 1. Given the vector function, find the integral. F(t)= (e2,4sin 2tarrow_forwardHow to use this integral ?arrow_forward
- Complex variablesarrow_forwardZero curl Consider the vector field y F i + sj+ zk. x? + y x? + y? a. Show that V × F = 0. b. Show that fF · dr is not zero on a circle C in the xy-plane enclosing the origin. c. Explain why Stokes' Theorem does not apply in this case.arrow_forwardmaths 1819arrow_forward
- Stokes's Theorem / The Curl Theorem 2. Consider the vector field F(x, y, 2) = (yz?,x,x + y) and the closed curve C: r(t) = (cos t, sin t,0) 0sts 27 Note that C is the unit circle in the xy-plane traced out counterclockwise (as viewed from above). Also let D be the unit disc in the xy-plane and let E be the 45-degree cone whose tip is at the point (-1,0,0) and whose boundary is the curve C. That is, D= { (x,y,2) | x² + y²<1 and z = 0 } E = { (x,y.2)| z = J + y? – 1 and – 1szs0} a) Graph C,D, and E so that we can see what's going on. Note that D would be input as just z = 0, cut off by C as its boundary. You should note that E lines up with C as its boundary also. b) Find V x F, the curl of F. You will use this below. Now our goal is to verify the Curl Theorem, and again, we'll do it twice. The Curl Theorem claims that fF- dr = |v x F) - as = v×F) - as c) First evaluate the leftmost expression directly, the line integral of F along the closed curve C. d) Next evaluate the middle…arrow_forwardSet-up the integral being asked in the problem. No need to evaluate. Show all solutions.arrow_forwardcalculate div(F) and curl(F). F = (xy, yz, y² – x³)arrow_forward
- Using 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_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_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
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