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Using Stokes's Theorem In Exercises 7-16, use Stokes’s Theorem to evaluate
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CALCULUS EARLY TRANSCENDENTAL FUNCTIONS
- 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_forwardSketch the plane curve represented by the vector-valued function and give the orientation of the curve. (8) = cos(6) +2 sin(6) O -2 y y ⓇE y yarrow_forwardV = 6y – 16x + 9z + V¢ Find the curl of vector V in Cartesian coordinates, o is Continuously differentiable. Find: V x Varrow_forward
- Use Stokes' Theorem to evaluate F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y?)i + (y + z?)j + (z + x²)k, C is the triangle with vertices (5, 0, 0), (0, 5, 0), and (0, 0, 5).arrow_forwardUse Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi +3zj + xk C: triangle with vertices (8, 0, 0), (0, 8, 0), (0, 0, 8)arrow_forwardProve the following Stokes theorem. (vector analysis)arrow_forward
- Calculate the directional derivative of g(x, y, z) = x In (y + z) in the direction v = 5i – 4j + 4k at the point P = (8, e, e). Remember to use a unit vector in directional derivative computation. (Use symbolic notation and fractions where needed.) Dyg(8, e, e) =arrow_forwardFind r(t) u(t). r(t). u(t) r(t) = (4 cos(t), 9 sin(t), t – 4), u(t) = (18 sin(t), −8 cos(t), t²) Is the result a vector-valued function? Explain. Yes, the dot product is a vector-valued function. No, the dot product is a scalar-valued function.arrow_forwardRepresent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.) P(0, 0, 0), Q(4, 2, 4) r(t) = %3D Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)arrow_forward
- (d) Use Stokes' theorem to evaluate f. F.dr where F = z?î + y?j + xk and C is the %3D triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1). The unit vector normal is upward.arrow_forwardVector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector normal to the surface S. Determine F as well as the area of the plane perpendicular to F if surface S = 2xy + 3z.arrow_forwardSketch the curve represented by the vector-valued function r(t) = 2 cos ti + tj + 2 sin tk and give the orientation of the curve.arrow_forward
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