Concept explainers
Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point (u, v) to the area of the image of that region near the point (x,y).
a. Suppose S is a rectangle in the uv-plane with vertices O(0,0), P(Δu, 0), {Δu, Δv), and Q(0, Δv) (see figure). The image of S under the transformation x = g(u, v), y = h(u, v) is a region R in the xy-plane. Let O’ P’ and Q’ be the images of O, P, and Q, respectively, in the xy-plane, where O’ P’ and Q’ do not all lie on the same line. Explain why the coordinates of O’, P’, and Q’ are (g(0, 0), h(0, 0)), (g(Δu, 0), h(Δu, 0)), and (g(0, Δv), h(0, Δv)), respectively.
b. Use a Taylor series in both variables to show that
where gu (0,0) is
c. Consider the
d. Explain why the ratio of the area of R to the area of S is approximately |J(u, v)|.
Want to see the full answer?
Check out a sample textbook solutionChapter 16 Solutions
CALCULUS: EARLY TRANSCENDENTALS (LCPO)
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Precalculus
Precalculus Enhanced with Graphing Utilities (7th Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,