Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S,
where N is the upward unit normal
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Chapter 15 Solutions
Calculus, Early Transcendentals
- An exercise on the gradient of a vector field Consider a potential function of the form • U(x, y) = Ax² + Bxy + Cy² + Dx + Ey+F Compute the gradient vector VU (x, y). Answer: U(x, y) = (2Ax+By+D,Bx+2C y +E) ⚫ Pick some values for A, B, C, D, E, F out of a hat (keep it simple!) • Ask yourself: does there exist a point (x, y) at which the gradient vector VU(x, y) is the zero vector? If so, is that point unique? • Repeat as necessary. • What conditions on A, B, C, D, E, F are necessary and sufficient for the existence of a point (x, y) at which VU (x, y) is the zero vector? If that point exists, is it unique?arrow_forwardFlux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?arrow_forwarddouble check work plsarrow_forward
- Subject differential geometry Let X(u,v)=(vcosu,vsinu,u) be the coordinate patch of a surface of M. A) find a normal and tangent vector field of M on patch X B) q=(1,0,1) is the point on this patch?why? C) find the tangent plane of the TpM at the point p=(0,0,0) of Marrow_forwardFind the curl of the vector field F(x, y, z) = x²i – 3j + yz?k. curl F = z?k curl F = z2j curl F = z2; curl F = z2i - jarrow_forwardLinear Algebra question is attached.arrow_forward
- xarrow_forwardCompute the flux of the vector field F(x, y, z)-6i+6j+ 4k through the rectangular region with corners at (1,0,0). (1,1,0), (1,1,2), and (1,0,2) oriented in the positive x direction, as shown in the figure. Flux y/7215 Z (Drag to rotate)arrow_forwardFlux of a vector field? Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.arrow_forward
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