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
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- Gravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.arrow_forwardEXERCISE 5 The temperature is T degrees at any point (x, y, z) in three-dimensional space and T(x,y, z) = 1/(x² + y² + z² + 3). %3D Distance is measured in inches. (a) Find the rate of change of the temperature at the point (3, –2, 2) in the direction of the vector –2 i+3j-6k. (6) Find the direction and magnitude of the greatest rate of change of T at (3,–2,2). 14arrow_forwardSketch some vectors in the vector field F(x, y) = 2xi + yj.arrow_forward
- Using Green's Theorem, find the outward flux of F across the closed curve C. F= (x- y)i + (x+ y)j; C is the triangle with vertices at (0,0), (10,0), and (0,5) O A. 0 В. 250 ОС. 100 D. 50arrow_forwardCompute the flux of the vector field F(x, y, z) = 3i +2j+ 2k 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 =arrow_forwardLet V={f: R→R: fis continuous}. V forms a vector space under the "usual" addition and scalar multiplication. What is the zero vector (ie, additive identity) of this vector space?arrow_forward
- Compute the flux of the vector field F(x, y, z) = 3i + 2j + 2k 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 = (Drag to rotate)arrow_forwardSketch the vector field. F(x, y) =〈y, 1〉arrow_forwardConsider the vector-valued function r(t) = cos ti + sin tj + In(cos t)k. (a) Find the vectors T, N, and B of r at the point P = (1,0,0). (b) Find the tangential and normal components of the acceler- ation of r at the point P = (1,0,0).arrow_forward
- Compute the flux of the vector field F(x,y, z) = 6i + 5j+3k through the rectangular region with corners at (0,0,1), (1,0, 1), (1,1,1), and (0,1,1) oriented in the positive z- direction, as shown in the figure. 1.50 Flux = $0.79S5 (Drag to rotate)arrow_forwardFind the gradient vector field (F(r, y, z)) of f(r, y, z) = y cos F(z, y, 2) = %3D Question Help: D Video Message instructor D Post to forum Add Work Submit Questionarrow_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_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage