Chapter 2: Problem 1

Compute the first fundamental forms of the following parametrized surfaceswhere they are regular: a. \(\mathbf{x}(u, v)=(a \sin u \cos v, b \sin u \sin v, c \cos u) ;\)ellipsoid. b. \(\mathbf{x}(u, v)=\left(a u \cos v, b u \sin v, u^{2}\right)\); ellipticparaboloid. c. \(\mathbf{x}(u, v)=\left(a u \cosh v, b u \sinh v, u^{2}\right)\); hyperbolicparaboloid. d. \(\mathbf{x}(u, v)=(a \sinh u \cos v, b \sinh u \sin v, c \cosh u)\);hyperboloid of two sheets.

### Short Answer

Expert verified

The first fundamental forms are: (a) E = a^2 \cos^2 u \cos^2 v + b^2 \cos^2 u \sin^2 v + c^2 \sin^2 u, F = 0, G = a^2 \sin^2 u \sin^2 v + b^2 \sin^2 u \cos^2 v. (b) E = a^2 \cos^2 v + b^2 \sin^2 v + 4u^2, F = 0, G = a^2 u^2 \sin^2 v + b^2 u^2 \cos^2 v. (c) E = a^2 \cosh^2 v + b^2 \sinh^2 v + 4u^2, F = 0, G = a^2 u^2 \sinh^2 v + b^2 u^2 \cosh^2 v. (d) E = a^2 \cosh^2 u \cos^2 v + b^2 \cosh^2 u \sin^2 v + c^2 \sinh^2 u, F = 0, G = a^2 \sinh^2 u \sin^2 v + b^2 \sinh^2 u \cos^2 v.

## Step by step solution

01

## - Compute Tangent Vectors for Ellipsoid

Compute the partial derivatives of \( \textbf{x}(u, v) = (a \sin u \cos v, b \sin u \sin v, c \cos u) \) with respect to u and v. \[ \frac{\partial \textbf{x}}{\partial u} = (a \cos u \cos v, b \cos u \sin v, -c \sin u) \] \[ \frac{\partial \textbf{x}}{\partial v} = (-a \sin u \sin v, b \sin u \cos v, 0) \]

02

## - Compute the First Fundamental Form for Ellipsoid

Find the coefficients E, F, and G: \[ E = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial u}\right) = a^2 \cos^2 u \cos^2 v + b^2 \cos^2 u \sin^2 v + c^2 \sin^2 u \] \[ F = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = 0 \] \[ G = \left(\frac{\partial \textbf{x}}{\partial v} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = a^2 \sin^2 u \sin^2 v + b^2 \sin^2 u \cos^2 v \]

03

## - Compute Tangent Vectors for Elliptic Paraboloid

Compute the partial derivatives of \( \textbf{x}(u, v) = (a u \cos v, b u \sin v, u^2) \) with respect to u and v. \[ \frac{\partial \textbf{x}}{\partial u} = (a \cos v, b \sin v, 2u) \] \[ \frac{\partial \textbf{x}}{\partial v} = (-a u \sin v, b u \cos v, 0) \]

04

## - Compute the First Fundamental Form for Elliptic Paraboloid

Find the coefficients E, F, and G: \[ E = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial u}\right) = a^2 \cos^2 v + b^2 \sin^2 v + 4u^2 \] \[ F = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = 0 \] \[ G = \left(\frac{\partial \textbf{x}}{\partial v} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = a^2 u^2 \sin^2 v + b^2 u^2 \cos^2 v \]

05

## - Compute Tangent Vectors for Hyperbolic Paraboloid

Compute the partial derivatives of \( \textbf{x}(u, v) = (a u \cosh v, b u \sinh v, u^2) \) with respect to u and v. \[ \frac{\partial \textbf{x}}{\partial u} = (a \cosh v, b \sinh v, 2u) \] \[ \frac{\partial \textbf{x}}{\partial v} = (a u \sinh v, b u \cosh v, 0) \]

06

## - Compute the First Fundamental Form for Hyperbolic Paraboloid

Find the coefficients E, F, and G: \[ E = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial u}\right) = a^2 \cosh^2 v + b^2 \sinh^2 v + 4u^2 \] \[ F = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = 0 \] \[ G = \left(\frac{\partial \textbf{x}}{\partial v} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = a^2 u^2 \sinh^2 v + b^2 u^2 \cosh^2 v \]

07

## - Compute Tangent Vectors for Hyperboloid of Two Sheets

Compute the partial derivatives of \( \textbf{x}(u, v) = (a \sinh u \cos v, b \sinh u \sin v, c \cosh u) \) with respect to u and v. \[ \frac{\partial \textbf{x}}{\partial u} = (a \cosh u \cos v, b \cosh u \sin v, c \sinh u) \] \[ \frac{\partial \textbf{x}}{\partial v} = (-a \sinh u \sin v, b \sinh u \cos v, 0) \]

08

## - Compute the First Fundamental Form for Hyperboloid of Two Sheets

Find the coefficients E, F, and G: \[ E = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial u}\right) = a^2 \cosh^2 u \cos^2 v + b^2 \cosh^2 u \sin^2 v + c^2 \sinh^2 u \] \[ F = \left(\frac{\partial \textbf{x}}{\partial u} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = 0 \] \[ G = \left(\frac{\partial \textbf{x}}{\partial v} \cdot \frac{\partial \textbf{x}}{\partial v}\right) = a^2 \sinh^2 u \sin^2 v + b^2 \sinh^2 u \cos^2 v \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### parametrized surfaces

A parametrized surface is a mathematical representation of a surface in three-dimensional space. It is expressed as a vector function \(\textbf{x}(u, v) \), where \((u, v)\) are parameters from a region in the plane. The parameters determine the position on the surface. This concept helps bridge the gap between abstract mathematical definitions and real-world shapes.

Common examples include:

- Ellipsoid
- Elliptic paraboloid
- Hyperbolic paraboloid
- Hyperboloid of two sheets

Understanding parametrized surfaces is crucial. They form the basis for calculating various properties like the first fundamental form, which tells us about the geometry of the surface.

###### tangent vectors

Tangent vectors describe the directions on a surface. They are derived from partial derivatives of the parametrization function with respect to each parameter.

For a given parametrized surface \(\textbf{x}(u, v)\), the tangent vectors are:

- \( \frac{\frac{\textbf{x}}{\partial u} = \frac{ \textbf{x}(u, v) \)}
- \( \frac{\frac{\textbf{x}}{\partial v} = \frac{ \textbf{x}(u, v) \)}

Each of these vectors lies in the tangent plane to the surface at a given point. They are essential for calculating the first fundamental form and understanding the shape and orientation of the surface.

The vectors also help in finding gradients, curvature, and other properties.

###### ellipsoid

An ellipsoid is a three-dimensional surface that looks like a stretched or squished sphere. Its general parametrization is:

\(\textbf{x}(u, v) = (a \sin u \cos v, b \sin u \sin v, c \cos u) \)

Here, a, b, and c are constants that define the semi-axes of the ellipsoid.

The tangent vectors are:

- \( \frac{\frac{\textbf{x}}{\partial u}} = (a \cos u \cos v, b \cos u \sin v, -c \sin u) \)
- \( \frac{\frac{\textbf{x}}{\partial v}} = (-a \sin u \sin v, b \sin u \cos v, 0) \)

The first fundamental form coefficients are:

- \( E = a^2 \cos^2 u \cos^2 v + b^2 \cos^2 u \sin^2 v + c^2 \sin^2 u \)
- \( F = 0 \)
- \(G = a^2 \sin^2 u \sin^2 v + b^2 \sin^2 u \cos^2 v \)

###### elliptic paraboloid

An elliptic paraboloid resembles a parabola rotated around its axis, extending infinitely. It is parametrized as:

\(\textbf{x}(u, v) = (a u \cos v, b u \sin v, u^2) \)

Here, a and b determine the stretching along the x and y axes.

The tangent vectors are:

- \( \frac{\frac{\textbf{x}}{\partial u}} = (a \cos v, b \sin v, 2u) \)
- \( \frac{\frac{\textbf{x}}{\partial v}} = (-a u \sin v, b u \cos v, 0) \)

The first fundamental form coefficients are:

- \( E = a^2 \cos^2 v + b^2 \sin^2 v + 4u^2 \)
- \( F = 0 \)
- \(G = a^2 u^2 \sin^2 v + b^2 u^2 \cos^2 v \)

###### hyperbolic paraboloid

A hyperbolic paraboloid looks like a saddle, curving upwards in one direction and downwards in another. Its parametrization is:

\(\textbf{x}(u, v) = (a u \cosh v, b u \sinh v, u^2) \)

Here, a and b are constants affecting its shape.

The tangent vectors are:

- \( \frac{\frac{\textbf{x}}{\partial u}} = (a \cosh v, b \sinh v, 2u) \)
- \( \frac{\frac{\textbf{x}}{\partial v}} = (a u \sinh v, b u \cosh v, 0) \)

The first fundamental form coefficients are:

- \( E = a^2 \cosh^2 v + b^2 \sinh^2 v + 4u^2 \)
- \( F = 0 \)
- \(G = a^2 u^2 \sinh^2 v + b^2 u^2 \cosh^2 v \)

###### hyperboloid of two sheets

A hyperboloid of two sheets consists of two separate surfaces that curve back-to-back similarly to two dishes. The parametrization is:

\(\textbf{x}(u, v) = (a \sinh u \cos v, b \sinh u \sin v, c \cosh u) \)

Here, a, b, and c determine the shape's dimensions.

The tangent vectors are:

- \( \frac{\frac{\textbf{x}}{\partial u}} = (a \cosh u \cos v, b \cosh u \sin v, c \sinh u) \)
- \( \frac{\frac{\textbf{x}}{\partial v}} = (-a \sinh u \sin v, b \sinh u \cos v, 0) \)

The first fundamental form coefficients are:

- \( E = a^2 \cosh^2 u \cos^2 v + b^2 \cosh^2 u \sin^2 v + c^2 \sinh^2 u \)
- \( F = 0 \)
- \(G = a^2 \sinh^2 u \sin^2 v + b^2 \sinh^2 u \cos^2 v \)

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