Problem 3. Calculate the first fundamental forms of the following surfaces:(а) x(и, 0) — (и - v, и + v, и? + 0?).(b) x(u, v) = (cosh u, sinh u, v).Problem 4. Compute the coefficients of the first fundamental form, the determinantthe matrix [gijl, and the unit normal vector n for the following surfaces:of(a) Surface given by z =z(x, y).(b) Sphere given by x =(a cos 0 cos o, a sin 0 cos ø, a sin ø).(c) Cylinder x² + y?1 (you can use parametrization x =(cos 0, sin 0, z).

According to guidelines we should have been able to solve only first question at a time #3

Answered: Problem 3. Calculate the first…

Problem 3. Calculate the first fundamental forms of the following surfaces: (a) x(u, v) = (u - v, u + v, u² + v²). %3D (b) x(u, v) = (cosh u, sinh u, v).

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

Problem 3. Calculate the first fundamental forms of the following surfaces:
(а) x(и, 0) — (и - v, и + v, и? + 0?).
(b) x(u, v) = (cosh u, sinh u, v).
Problem 4. Compute the coefficients of the first fundamental form, the determinant
the matrix [gijl, and the unit normal vector n for the following surfaces:
of
(a) Surface given by z =
z(x, y).
(b) Sphere given by x =
(a cos 0 cos o, a sin 0 cos ø, a sin ø).
(c) Cylinder x² + y?
1 (you can use parametrization x =
(cos 0, sin 0, z).

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