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).
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).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 6AEXP
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![Problem 3. Calculate the first fundamental forms of the following surfaces:
(а) x(и, v) — (и - v, и + 0, и? + и?).
(b) x(u, v) = (cosh u, sinh u, v).
%3D
Problem 4. Compute the coefficients of the first fundamental form, the determinant
the matrix [gij], 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 + y² = 1 (you can use parametrization x =
(cos 0, sin 0, 2)).
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0124cc3-8e61-42ea-bb4b-64e63e0cb960%2F3b307eff-c6e2-40d4-a5af-406ee0048a80%2F2lxk34_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3. Calculate the first fundamental forms of the following surfaces:
(а) x(и, v) — (и - v, и + 0, и? + и?).
(b) x(u, v) = (cosh u, sinh u, v).
%3D
Problem 4. Compute the coefficients of the first fundamental form, the determinant
the matrix [gij], 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 + y² = 1 (you can use parametrization x =
(cos 0, sin 0, 2)).
%3D
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