Stokes’ Theorem on closed surfaces Prove that if F satisfies the conditions of Stokes’ Theorem, then
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- Verify Stokes's Theorem for F = z²î+ x²j + y²k and S is the surface z2 = x2 + y2, y 2 0, and 0arrow_forwardClairaut's Theorem Let DCR? be a disk containing the origin and assume that g : D R is a function given by g(x, y) = e" (cos y +r sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0, 0).arrow_forwardClairaut's Theorem Let DCR be a disk containing the origin and assume that q : D → R is a function given by g(x, y) = e" (cos y +x sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0, 0).arrow_forwardClairaut's Theorem Let DC R?be a disk containing the origin and assume g : D → R is a function given by that g(x,y) = e" (cos y + xsin y). Prove that g(x,y) satisfies the Clairaut Theorem at point (0,0).arrow_forwardClairaut's Theorem Let DC R? be a disk containing the origin and assume that g : D→ R is a function given by g(x, y) = e" (cos y + x sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0,0).arrow_forwardUse Stokes’ Theorem to evaluate ∫ F*dr where C is oriented counter-clockwise as viewed from above. F(x,y,z) = yi-zj+x2k C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1) Note: The triangle is a portion of the plane x+y+z=1arrow_forwardUse Stokes' Theorem to evaluate Use Stokes' Theorem to evaluate ∫C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + 3xzj + exyk, C is the circle x2 + y2 = 4, z = 6.arrow_forwardSketch the surface of f(x,y)arrow_forwardHow do you calculate the area of a parametrized surface in space? Of an implicitly defined surface F(x, y, z) = 0? Of the surface which is the graph of z = ƒ(x, y)? Give examples.arrow_forwardaut's Theorem Let DC R? be a disk containing the origin and assume that g : D → R is a function given by g(x, y) = e" (cos y + x sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0, 0).arrow_forward(3) Use Stokes' Theorem to evaluate | F. dr where F(x, y, z)æzî + 2ry} + 3ryk where C is the boundary of the part of the plane 3x + y + z = 3 in the first octant and C is oriented counterclockwise as viewed from above.arrow_forwardUse Stokes' Theorem to evaluate ∫ C F · dr where F = (x + 5z) i + (7x + y) j + (2y − z) k and C is the curve of intersection of the plane x + 3y + z = 12 with the coordinate planes.(Assume that C is oriented counterclockwise as viewed from above.)arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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