Day: March 14, 2023

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202303141200 Exercise 7.7.1

Consider the coordinate transformation (x,y)=G(u,v) given by

G(u,v)=(au,bv) for (u,v)\in\mathbb{R}^2

where a and b are positive constants.

(a) Prove that G is \textrm{1--1} on \mathbb{R}^2.
(b) Show that G maps the circular disc

K=\{(u,v)\in\mathbb{R}^2 : u^2+v^2\leqslant 1\}

in the u, v plane onto the elliptical disc

G(K)=\{(x,y)\in\mathbb{R}^2 : x^2/a^2+y^2/b^2\leqslant 1\}

in the x, y plane.
(c) Calculate \mathrm{det}\,J_{G,(u,v)}.
(d) Calculate, using Theorem 7.7.5, the area of the elliptical disc G(K). (Assume known that the area of K is \pi.)

Extracted from P. R. Baxandall. (1986). Vector Calculus.

Roughwork.

Symbolically,

\forall\, a,b\in X,\quad f(a)=f(b)\Rightarrow a=b,

or the contrapositive

\forall\, a,b\in X,\quad a\neq b\Rightarrow f(a)\neq f(b).

Wikipedia on Injective function

(a)

From G:\mathbb{R}^2\to\mathbb{R}^2:(u,v)\mapsto (x=au,y=bv),

\begin{aligned} & \enspace & (x_1,y_1)=G(u_1,v_1) & = G(u_2,v_2)=(x_2,y_2) \\ \Rightarrow & \enspace & (au_1,bv_1) & = (au_2,bv_2) \\ \Rightarrow & \enspace & (u_1,v_1) & = (u_2,v_2) \\ \end{aligned}

(b) Obvious.

(c)

\begin{aligned} J_{G}(u,v) & = \begin{bmatrix} \partial_u(au)&\partial_v(au) \\ \partial_u(bv) & \partial_v(bv) \\ \end{bmatrix} \\ & = \begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix} \\ \mathrm{det}\,J_{G,(u,v)} & = ab \\ \end{aligned}

(d)

Theorem (Change of variables).  Let G:K\subseteq \mathbb{R}^2\to\mathbb{R}^2 be a C^1 function defined on a compact set K in \mathbb{R}^2, and let D\subseteq K be an open subset of \mathbb{R}^2 such that

i. K\backslash D is a null set,
ii. G is \textrm{1--1} on D,
iii. \mathrm{det}\,J_{G,(u,v)}\neq 0 for all (u,v)\in D.

Then, for any bounded function f:G(K)\subseteq\mathbb{R}^2\to\mathbb{R} which is continuous on G(D)

\displaystyle{\iint_{G(K)}f(x,y)\,\mathrm{d}x\,\mathrm{d}y=\iint_{K}f(G(u,v))|\mathrm{det}\,J_{G,(u,v)}|\,\mathrm{d}u\,\mathrm{d}v}.

See Theorem 7.7.5 on pg. 404.

The area of the elliptical disc G(K) is \pi ab.

This problem is not to be attempted.

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202303141137 Exercise 2.7.1

Calculate the lengths of the following smooth simple arcs in \mathbb{R}^3.

(a) The circular helix parametrized by

f(t)=(t,\cos t,\sin t)

where t\in [a,b];
(b) the curve parametrized by

f(t)=(e^t\cos t,e^t\sin t,e^t)

where t\in [0,k].

Extracted from P. R. Baxandall. (1986). Vector Calculus.

Roughwork.

Let f:[a,b]\subseteq\mathbb{R}\to\mathbb{R}^n be a C^1 path in \mathbb{R}^n. The length of f is defined to be

l(f)=\displaystyle{\int_{a}^{b}\|f'(t)\|\,\mathrm{d}t}.

See Definition 2.7.2 on pg.60

(a)

\begin{aligned} f(t) & = (t,\cos t,\sin t) \\ f'(t) & = (1,-\sin t,\cos t) \\ \|f'(t)\| & = \sqrt{(1)^2+(-\sin t)^2+(\cos t)^2} \\ & = \sqrt{2} \\ l(f) & = \int_{a}^{b} \sqrt{2}\,\mathrm{d}t \\ & = \sqrt{2}(b-a) \\ \end{aligned}

(b) Not to be attempted.