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Chapter 15 Solutions
Calculus (MindTap Course List)
- Properties of div and curl Prove the following properties of thedivergence and curl. Assume F and G are differentiable vectorfields and c is a real number.a. ∇ ⋅ (F + G) = ∇ ⋅ F + ∇ ⋅ Gb. ∇ x (F + G) = (∇ x F) + (∇ x G)c. ∇ ⋅ (cF) = c(∇ ⋅ F)d. ∇ x (cF) = c(∇ ⋅ F)arrow_forwardSketch some vectors in the vector field F(x, y) = −yi + xj.arrow_forwardRain 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_forward
- Represent the line segment from P(0, 2, −7) to Q(4, 7, 1) by a vector-valued function (use the standard unit vector notation) r(t) = 0 ≤ t ≤ 1, and also by a set of parametric equations X Y N ||arrow_forwardSketch the vector field F. O O -2 F(x, y) = i + (y − x)j + 1 X 2 Xarrow_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
- Sketch the vector field. x|y|F(x, y) = = 0 -4 2 02 -2 0 -2 2 2 -2 0 -3 -2 2 -2 -2 № → 3 2 1 -2 ین Y " X 2 1 2 3 +arrow_forwardVV by Consider the vector field F(x, y, z) = (6y, 6x, — z). Show that F is a gradient vector field F = determining the function V which satisfies V(0, 0, 0) = 0. V(x, y, z)arrow_forwardExample Let F = xy? i+ xy j be a vector field in 2-space. Evaluate $. xy? dx + xy? dy where C is the boundary of the triangle with vertices (0,2),(3,2), and (3,5). (3,5) y+2 (0,2) (3,2) y=2 Example Let C be the curve sketched below and F(x,y, 2) = 3xy i+ 3zj+ 5x R. The straight line on the xy-plane is given by the equation 2x + 3y = 6 and the curve on the yz-plane has an equation of z= 4- y?. Find S. F dř. (00.4) (02,0) (3,0,0), 2x+3y=6arrow_forward
- U (x, y) 4c(y + 1)i + xyj, and V(x, y) = Consider two vector fields in the xy plane, given in the Cartesian coordinates as: cyi - xj, where c is a constant. Find where in the xy plane the vectors of these two fields are parallel to one another, and where they are mutually orthogonal. =arrow_forwardQuestion: Prove that the 2d-curl of a conservative vector field is zero, ( ∇ × ∇ f ) ⋅ k = 0 (here k is unit vector) for any general scalar function f ( x , y ).arrow_forwardSketch the vector fields. Use a table for it. F(x,y)=<x,y-x>arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning