Curl of a Cross Product In Exercises 69 and 70, find
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Calculus
- Nonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?arrow_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_forwardApplication of Green's theorem Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that JS D Ur Vy dA= u dv, where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forward
- Find r(t)u(t). r(t) = (3-7)i + +³1 + u(t) = t²i - 8j + t³k r(t)u(t)= j+ 4k Is the result a vector-valued function? Explain. Need Help? Yes, the dot product is a vector-valued function. O No, the dot product is a scalar-valued function. Watch Itarrow_forwardVector Calculus 1) Find the directional derivatives as a shown function of f at P (1,2,3) in the direction from P to Q (4,5,2) f(x, y, z) = x³y – yz² + zarrow_forwardCalculate the line integral of the vector field F = (y, x,x² + y² ) around the boundary curve, the curl of the vector field, and the surface integral of the curl of the vector field. The surface S is the upper hemisphere x² + y + z? = 25, z 2 0 oriented with an upward-pointing normal. (Use symbolic notation and fractions where needed.) F. dr = curl(F) =arrow_forward
- Differentiation of Vector-Valued Functions In Exercises 7 and 8, find r (t), r(t,), and r (t,) for the given value of t. Then sketch the space curve represented by the vector-valued function, and sketch the vectors r(t) and r'(t). 7. r(t) = 2 cos ti + 2 sin tj + tk, toarrow_forwardfind curl (curl F) = V x (V X F).arrow_forwardGreen's Second Identity Prove Green's Second Identity for scalar-valued functions u and v defined on a region D: (uv²v – vv²u) dV = || (uvv – vVu) •n dS. (Hint: Reverse the roles of u and v in Green's First Identity.)arrow_forward
- Rain on a roof Consider the vertical vector field F = ⟨0, 0, -1⟩, correspondingto a constant downward flow. Find the flux in the downward direction acrossthe surface S, which is the plane z = 4 - 2x - y in the first octant.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_forwardFlux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.arrow_forward
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