F(x,y,z)=xyi+x^2j+z^2k;C is the intersection of the paraboloid z = x^2 + y^2 and

Question

F(x,y,z)=xyi+x^2j+z^2k;C is the intersection of the paraboloid z = x^2 + y^2 and the plane z = y with a counter- clockwise orientation looking down the positive z-axis. Please use the left hand side of stokes theorem. the parameterization.

Explanation / Answer

Let S be the surface inside by C. It is enclosed by x^2 + y^2 = z = y. ==> x^2 + (y - 1/2)^2 = 1/4, by completing the square. (This is a circle with radius 1/2.) So, ?c F · dr = ??s curl F · dS, by Stokes' Theorem = ?? · dA, since C lies on the plane z = y = ?? · dA = ?? x dA Now, convert to polar coordinates. x^2 + y^2 = y ==> r^2 = r sin ?. ==> r = sin ?, which is traversed for ? in [0, p]. So, the integral equals ?(? = 0 to p) ?(r = 0 to sin ?) (r cos ?) * (r dr d?) = ?(? = 0 to p) (1/3)r^3 cos ? {for r = 0 to sin ?} d? = ?(? = 0 to p) (1/3) sin^3(?) cos ? d? = (1/12) sin^4(?) {for ? = 0 to p} = 0.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.