Concept explainers
Evaluating a Line Integral In Exercises 23-32, evaluate
along each path. (Hint: If F is conservative, the
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CALCULUS EARLY TRANSCENDENTAL FUNCTIONS
- Let f = f(x, y, z) be a sufficiently smooth scalar function and F = Vƒ be the gradient acting on f. Which of the following expressions are meaningful? Of those that are, which are necessarily zero? Show your detailed justifications. (a) V· (Vf) (b) V(V × f) (c) V × (V · F) (d) V. (V × F)arrow_forwardConsider the vectorial V=2+ŷ+ 2 z . . function=z²+x²y + y²2 and the gradient operator Please explicitly evaluate Vxarrow_forwardEvaluate f**dx A. 를 (1+)를 -를 (1+ 즐)를 + C C. + C D. - (1-5 + c в. + C Evaluate the integral S(+x)° dx. A. +2 In In x + 3x? –+ C c. -+ 2 In lIn x – 3x² ++ C 4x4 x* + C 3x2 3x2 x* - 3 In In x +: 2x D. -+3 In In x ++÷+c B. Find the area enclosed by the curve r?=10c cos (20) A. 5 sg. units B. 10 sq. units C. 15 sq. units 20 sq. units D.arrow_forward
- Need helparrow_forwardmaths 1819arrow_forward3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.arrow_forward
- Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (−4, 2), (−4, −3), (2, −2), (2, 7), and back to (−4, 2), in that order. Use Green's theorem to evaluate the following integral. √ (2xY) (2xy) dx + (xy2) dyarrow_forwardUse Green's Theorem to evaluate the line integral § 2y² dx + 3x² dy, where C is the boundary of the square -1 ≤ x ≤ 1, -1 ≤ y ≤ 1. Orient the curve counterclockwise. (Use symbolic notation and fractions where needed.) 2y² dx + 3x² dy =arrow_forwardDetermine whether the line integral of each vector field (in blue) along the semicircular, oriented path (in red) is positive, negative, or zero. Positive Positive Zero Zero Negative Positive - 1.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,