Let $R$ be the rectangle whose vertices are $(1,1), (1 , 3/2), (3/2 , 1), (3/2 , 3/2).$ Then...

Question

Let $$ [x,y] = T[u,v]=[u]=[u^2-v^2,2uv] \\$$

1. Show that T is injective on R.
2. Sketch the square image by T.
3. Sketch thequare image by $$ \left.dT\right|_{(1,1)}(u,v)$$ This image is a square?
4. Show that translation of the image found by vector $$ T(1,1)- \left.dT\right|_{(1,1)}(1,1) $$ is the square image by linear approximation $$A(uv)$$ to near $(1,1)$.

I tried to do and I did not succeed. If you can help me with any items, I'll be grateful.

Answer 1

I understand that the question is about the map $$T:\quad(u,v)\mapsto (x,y)=T(u,v):=(u^2-v^2,2uv)\ ,$$ restricted to the given square $Q$.

${\bf 1\ }$ Assume we have two points $(u_i, v_i)\in Q$ with $T(u_1,v_1)=T(u_2,v_2)=:(x,y)$. Then $$x^2+y^2=(u_i^2-v_i^2)^2+4u_i^2v_i^2=(u_i^2+v_i^2)^2\qquad(i=1, 2)\ .$$ This allows to conclude that $u_1^2+v_1^2=u_2^2+v_2^2$, and together with $u_1^2-v_1^2=x=u_2^2-v_2^2$ it follows that $u_1^2=u_2^2$, $v_1^2=v_2^2$. Since $Q$ is in the first quadrant we may take square roots here, and obtain $(u_1,v_1)=(u_2,v_2)$.

${\bf 2\ }$ The following picture incorporates also part ${\bf 4}$:

${\bf 3\ }$ The Jacobian is $$dT(u,v)=\left[\matrix{2u&-2v\cr 2v&2u\cr}\right]\ ,$$ hence $$dT(1,1)=\left[\matrix{2&-2\cr 2&2\cr}\right]=\sqrt{8}\left[\matrix{{1\over\sqrt{2}}&{-1\over\sqrt{2}}\cr {1\over\sqrt{2}}&{1\over\sqrt{2}}\cr}\right]\ .$$ The last matrix describes a counterclockwise rotation by $45^\circ$, followed by a scaling. Therefore it transforms squares into squares.

${\bf 4\ }$ The map $$(u,v)\mapsto T(1,1)+dT(1,1).(u-1,v-1)$$ is an approximation to $T$ in the neighborhood of $(1,1)$ and maps $Q$ to some euclidean square $Q'$ which "approximates" the actual image $T(Q)$.