Investigation Consider the graph of the vector-valued function
(a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints.
(b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the
(c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b).
(d) Use the
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Chapter 12 Solutions
Multivariable Calculus
- Identities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ x (F x G ) = (G ⋅ ∇) F - G (∇ ⋅ F) - (F ⋅ ∇)G + F (∇ ⋅ G)arrow_forwardIdentities Prove the following identities. Assume φ is a differentiablescalar-valued function and F and G are differentiable vectorfields, all defined on a region of ℝ3. ∇ x (∇ x F) = ∇(∇ ⋅ F) - (∇ ⋅ ∇)Farrow_forwardCirculation on a half-annulus Consider the vector field F = ⟨y2, x2⟩on the half-annulus R = {(x, y): 1 ≤ x2 + y2 ≤ 9, y ≥ 0}, whose boundary is C. Find the circulation on C, assuming it has the orientation shown.arrow_forward
- Interpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forwardMaximum curl Let F = ⟨z, x, -y⟩.a. What is the scalar component of curl F in the direction of n = ⟨1, 0, 0⟩?b. What is the scalar component of curl F in the direction ofn = ⟨0, -1/√2, 1/√2⟩?c. In the direction of what unit vector n is the scalar componentof curl F a maximum?arrow_forwardUsing a Function (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f (x, y) = c at P, (c) find the tangent line to the level curve f (x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 + 4y2, c = 40, P(2, −1)arrow_forward
- Lines as vector-valued functions Find a vector function for the line thatpasses through the points P(2, -1, 4) and Q(3, 0, 6).arrow_forwardFlux Consider the vector fields and curve. a. Based on the picture, make a conjecture about whether the outwardflux of F across C is positive, negative, or zero.b. Compute the flux for the vector fields and curves. F and C givenarrow_forwardProperties 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_forward
- Integrals of Line and Work A cyclist rides up a mountain along the path shown in the figure. She makes one complete revolution around the mountain in reaching the top, while her vertical rate of climb is constant. Throughout the trip, she exerts a force described by the vector field F(x,y,z) = z2i + 3y2j + 2xk What is the work done by the cyclist in travelling from A to B?arrow_forwardVector A satisfies the vector transformation law, A'=SA. Show directly that its time derivative dA/dt also satisfies A'=SA and is therefore a vectorarrow_forwardVector-valued functions Find a function r(t) that describes the following curve. The line passing through the point P(4, -2, 3) that is orthogonalto the lines R(t) = ⟨t, 5t, 2t⟩ and S(t) = ⟨t + 1, -1, 3t - 1⟩arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning