Solved: TEAM PROJECT. First Fundamental Form of a Surface.

Chapter 10, Problem 10.1.134

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First Fundamental Form of a Surface. Given a surface \(S: \mathbf{r}(u, v)\), the corresponding quadratic differential form

(13)          \(d s^{2}=E d u^{2}+2 F d u d v+G d v^{2}\)

with coefficients

(14)          \(E=\mathbf{r}_{u} \cdot \mathbf{r}_{u}, \quad F=\mathbf{r}_{u} \cdot \mathbf{r}_{v}, \quad G=\mathbf{r}_{v} \cdot \mathbf{r}_{v}\)

is called the first fundamental form of S. (E, F, G are standard notations that have nothing to do with F and G that occur at some other places in this chapter.) The first fundamental form is basic in the theory of surfaces since with its help we can determine lengths, angles, and areas on S. To show this, prove the following.

(a) For a curve \(C: u=u(t), v=v(t), a \leqq t \leqq b\), on S, formulas (10), Sec. 9.5, and (14) give the length

\(l=\int_{a}^{b} \sqrt{\mathbf{r}^{\prime}(t) \cdot \mathbf{r}^{\prime}(t)} d t\)

(15)         \(=\int_{a}^{b} \sqrt{E u^{\prime 2}+2 F u^{\prime} v^{\prime}+G v^{\prime 2}} d t\)

(b) The angle \(\gamma\) between two intersecting curves \(C_{1}: u=g(t), v=h(t)\) and \)C_{2}: u=p(t), v=q(t)\) on \(S: \mathbf{r}(u, v)\) is obtained from

(16)          \(\cos \gamma=\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)

where \(\mathbf{a}=\mathbf{r}_{u} g^{\prime}+\mathbf{r}_{v} h^{\prime}\) and \(\mathbf{b}=\mathbf{r}_{u} p^{\prime}+\mathbf{r}_{v} q^{\prime}\) are tangent vectors of \(C_{1}\) and \(C_{2}\).

(c) The square of the length of the normal vector \(\mathbf{N}\) can be written

(17)          \(\quad|\mathbf{N}|^{2}=\left|\mathbf{r}_{u} \times \mathbf{r}_{v}\right|^{2}=E G-F^{2}\).

so that formula (8) for the area A(S) of S becomes

                 \(A(S) =\iint_{S} d A=\int_{R}|\mathbf{N}| d u d v\)

(18)

                 \(=\int_{R} \sqrt{E G-F^{2}} d u d v\)

(d) For polar coordinates u(=r) and \(v(=\theta)\) defined by x = u cos v, y = u sin v we have

E= 1, F = 0,  \(G=u^{2}\), so that

\(d s^{2}=d u^{2}+u^{2} d v^{2}=d r^{2}+r^{2} d \theta^{2}\).

Calculate from this and (18) the area of a disk of radius a.

(e) Find the first fundamental form of the torus in Example 5. Use it to calculate the area A of the torus. Show that A can also be obtained by the theorem of Pappus, \({ }^{7}\) which states that the area of a surface of revolution equals the product of the length of a meridian C and the length of the path of the center of gravity of C when C is rotated through the angle \(2 \pi\).

(f) Calculate the first fundamental form for the usual representations of important surfaces of your own choice (cylinder, cone, etc.) and apply them to the calculation of lengths and areas on these surfaces.  

Text Transcription:

S: r (u, v)

ds^2 = E du^2 + 2F du dv + G dv^2

E = r_u cdot r_u,     F = r_u cdot r_v,     G = r_v cdot r_v

C: u = u(t), v = v(t), a leqq t leqq b

I = int_{a}^{b} sqrt{r ‘(t) cdot r ‘ (t)} dt

= int_{a}^{b} sqrt{E u ‘ + 2Fu ‘  v ‘ + Gv’2} dt

C_1: u = g(t), v = h(t)

C_2: u = p(t), v = q(t)

S: r (u, v)

cos gamma = a cdot b / |a||b|

cos gamma = a cdot b / |a||b|

a = r_u g’ + r_v h’

b = r_u p’ + r_v q’

C_1

C_2

N

|N|^2 = |r_u X r_v|^2 = E G - F^2

A(S) = iint_{S} dA = int_R| N| du dv

= int_R sqrt{E G - F^2} du dv

v (= theta)

G = u^2

ds^2 = du^2 + u^2 dv^2 = dr^2 + r^2 d theta^2.

^7

2 pi