Concept explainers
Evaluating line
a. By parameterizing C
b. By using the Fundamental Theorem for line integrals, if possible
27. F = 〈y, z, –x〉; C: r(t) = 〈cos t, sin t, 4〉, for 0 ≤ t ≤ 2π
Trending nowThis is a popular solution!
Chapter 14 Solutions
Calculus: Early Transcendentals (2nd Edition)
Additional Math Textbook Solutions
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Calculus & Its Applications (14th Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Calculus and Its Applications (11th Edition)
University Calculus: Early Transcendentals (3rd Edition)
University Calculus: Early Transcendentals (4th Edition)
- Number 3. This is Calculus 3arrow_forward(3) Evaluate the integral F. dr where the vector F is defined by F = and where the curve C is defined by r(t) = for 0 ≤ t ≤ 2Tarrow_forward42. Derivatives of triple scalar products a. Show that if u, v, and w are differentiable vector functions of t, then du v X w + u• dt dv X w + u•v X dt dw (u•v X w) dt dt b. Show that d'r dr? dr dr d'r dt dt r. dt dr? (Hint: Differentiate on the left and look for vectors whose products are zero.)arrow_forward
- 1. Consider the function F(x, y, z) = (√/1 – x² − y², ln(e² — z²)). This function is a mapping from R" to Rm. Determine the values of m and n. (b) Is this function scalar-valued or vector-valued? Briefly explain. (c) Determine the domain and range of F and sketch the corresponding regions. (d) Is it possible to visualize this function as a graph? If so, sketch the graph of F.arrow_forwardProve that the tangent vector is always orthogonal tothe position vector for the vector-valued function r(t) =⟨sin t, sint cos t, cos2 t⟩.arrow_forwardLet f(x, y) = 7x cos (y). Find the conservative vector field F, which is the gradient of f. (Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.) F = (7 cos (y).-7x sin(y) Evaluate the line integral of F over the upper half of the unit circle centered at the origin, oriented clockwise. (Give an exact answer. Use symbolic notation and fractions where needed.) C F. dr = Incorrect 12 Question Source: Rogawski 4e Calculus Early Transcendentals Publisher: W.H. Freema 5.02. Darrow_forward
- Flux across curves in a vector field Consider the vector fieldF = ⟨y, x⟩ shown in the figure.a. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for 0 ≤ t ≤ π/2.b. Compute the outward flux across the quarter-circleC: r(t) = ⟨2 cos t, 2 sin t⟩ , for π/2 ≤ t ≤ π.c. Explain why the flux across the quarter-circle in the third quadrant equals the flux computed in part (a). d. Explain why the flux across the quarter-circle in the fourth quadrant equals the flux computed in part (b).e. What is the outward flux across the full circle?arrow_forwardShow that the vector-valued function shown below describes the function of a particle moving in a circle of radius 1 centered at a point (5,5,3) and lying in the plane 3x+3y-6z = 12arrow_forwardDetermine which theorem applies to simplify the following. It is possible that more than one will apply. Find the work done by the vector field F(r, y) = yzcos(ry)i + zzcos(ry)j+ sin(ry)k along the positively oriented triangle with vertices (2,1,-6), (3,-1,1), and (0,1,0). O Fundamental Theorem of Line Integrals O Green's Theorem O Stoke's Theorem OThe Divergence Theorem Question Help: Message instructor D Post to forum Add Work Submit Questionarrow_forward
- OSketch the vector field F(x, y) = -5i – 5j and calculate the line integral of F along the line segment from (3, –3) to (-4, –4).arrow_forwardHow do I do d) and e)? For d) I took the derivative of the vector r but I got a really weird solution which seems impossiblearrow_forwardFind the domain of the vector function. lim t->infinity <1+ t2/1- t2, tan-1(t), 1- e-2t/t>arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning