Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S,
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Calculus
- ٦arrow_forward#12arrow_forward(5) Let ß be the vector-valued function 3u ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = { 3u² 4 B (0,7), 0₁B (0,7), 0₂B (0,7) u cos(v) VI+ u², sin(v), (a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). (b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and varying u, and (ii) the path obtained by holding u = O and varying v. (c) Compute the following quantities: (d) Draw the following tangent vectors on your sketch in part (a): X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)* ' cos(v) √1+u² +arrow_forward
- Sketch the vector fields. Use a table for it. F(x,y)=<x,y-x>arrow_forwardHow do you graph the vector field F = ⟨ƒ(x, y), g(x, y)⟩?arrow_forwardSubject 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_forward
- solve a part onlyarrow_forwardShow that the vector-valued function shown below describes the function of a particle moving in a circle of radius 1 centered at a point (5,5,3) and lying in the plane 3x+3y-6z = 12arrow_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_forward
- D part onlyarrow_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 Find the integral of the vector function F(t)=(f.,cost)arrow_forwardAn 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_forward
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