Line
is the same for each parametric representation of C.
(i)
(ii)
Trending nowThis is a popular solution!
Chapter 15 Solutions
Multivariable Calculus (looseleaf)
- Sketch 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_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
- Sketch the vector field. F(x, y) =〈y, 1〉arrow_forwardLet ø = p(x), u = u(x), and T = T(x) be differentiable scalar, vector, and tensor fields, where x is the position vector. Show that %3Darrow_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
- Divergence and Curl of a vector field are Select one: a. Scalar & Scalar b. Non of them c. Vector & Scalar d. Vector & Vector e. Scalar & Vectorarrow_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_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_forward
- VV 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_forwardGreen's Theorem as a Fundamental Theorem of Calculus Show that if the circulation form of Green's f(x)' Theorem is applied to the vector field ( 0,). where c > 0 and R = {(x, y): a SIS b,0 s ys c}, then the result is the Fundamental Theorem of Calculus, dx = f(b) – f(a).arrow_forwardFind ||F||. F(x, y) = i + j Sketch several representative vectors in the vector fieldarrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning