Concept explainers
Let F(x, y) =
(a) Show that
(b) Show that ∫C F · dr is not independent of path. [Hint: Compute ∫
Trending nowThis is a popular solution!
Chapter 16 Solutions
Calculus, Early Transcendentals
- If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xyarrow_forwardLet C denote the circle of radius 1 in R2centered at the origin, oriented counterclockwise.for which F(x, y) = <1,y> Compute F*drarrow_forwardSuppose F(x,y)=(4x+6y)i +(6x+4y)j. Evaluate the line integral for each of the given paths, which are comprised of line segments and arcs of circles. (a) If CC is the open semicircular path in figure I of radius 33 oriented counterclockwise, then? (b) If CC is the open piecewise linear path in figure II from (3,0) to (0,−3) to (−3,0), then? (c) If CC is the closed path in figure III along the boundary of a semi-disk of radius 33 oriented counterclockwise, then?arrow_forward
- Let X=R2 and defined d2: R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a metric on R2arrow_forwardSuppose C is a curve of length N, and ||F|| ≤ M, where M is some positivenumber. Prove thatarrow_forwardCompute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 andC is part of circle x^2 + y^2 = 9 with x ≥0, y ≥0, oriented clockwise.arrow_forward
- Show that the tangent plane to the surface x2a2+y2b2= cz at the point P(x0, y0, z0) is given by theequation2xx0a2+2yy0b2= c(z + z0) .arrow_forwardLet F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate ∮C F • dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).arrow_forwardShow that f(x,y) = xy (a) satisfies a Lipschitz condition on any rectangle a ≤ x ≤ b and c ≤ y ≤ d; (b) satisfies a Lipschitz condition on any strip a ≤ x ≤ b and −∞ < y < ∞; (c) does not satisfy a Lipschitz condition on the entire plane.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,