변수변환과 야코비안

In the following exercises, use the transformation $u=x+y,v=x-y$ to evaluate the integrals on the trapezoidal region $R$ determined by the points $(1,0),(2,0),(0,2),\text{and}\,(0,1)$ shown in the following figure.

$\iint_{R}({x}^{2}-2xy+{y}^{2}){e}^{x+y}dA$

In the following exercises, use the transformation $u=x+y,v=x-y$ to evaluate the integrals on the trapezoidal region $R$ determined by the points $(1,0),(2,0),(0,2),\text{and}\,(0,1)$ shown in the following figure.

$\iint_{R}({x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3})dA$

The circular annulus sector $R$ bounded by the circles $4{x}^{2}+4{y}^{2}=1$ and $9{x}^{2}+9{y}^{2}=64,$ the line $x=y\sqrt{3},$ and the $y\text{-axis}$ is shown in the following figure. Find a transformation $T$ from a rectangular region $S$ in the $r\theta \text{-plane}$ to the region $R$ in the $xy\text{-plane.}$ Graph $S.$

The solid $R$ bounded by the circular cylinder ${x}^{2}+{y}^{2}=9$ and the planes $z=0,z=1,$ $x=0,\text{and}\,y=0$ is shown in the following figure. Find a transformation $T$ from a cylindrical box $S$ in $r\theta z\text{-space}$ to the solid $R$ in $xyz\text{-space}.$

Show that $\iint_{R}f\left(\sqrt{\frac{{x}^{2}}{3}+\frac{{y}^{2}}{3}}\right)dA=2\pi \sqrt{15}\int_{0}^{1}f(\rho )\rho \,d\rho ,$ where $f$ is a continuous function on $[0,1]$ and $R$ is the region bounded by the ellipse $5{x}^{2}+3{y}^{2}=15.$

Show that $\iiint_{R}f\left(\sqrt{16{x}^{2}+4{y}^{2}+{z}^{2}}\right)dV=\frac{\pi }{2}\int_{0}^{1}f(\rho ){\rho }^{2}d\rho ,$ where $f$ is a continuous function on $[0,1]$ and $R$ is the region bounded by the ellipsoid $16{x}^{2}+4{y}^{2}+{z}^{2}=1.$

[T] Find the area of the region bounded by the curves $xy=1,xy=3,y=2x,$ and $y=3x$ by using the transformation $u=xy$ and $v=\frac{y}{x}.$ Use a computer algebra system (CAS) to graph the boundary curves of the region $R.$

[T] Find the area of the region bounded by the curves $xy=2,xy=3,y=x,$ and $y=2x$ by using the transformation $u=xy$ and $v=\frac{y}{x}.$

Evaluate the triple integral $\int_{0}^{1}\,\int_{1}^{2}\,\int_{z}^{z+1}(y+1)dx\,dy\,dz$ by using the transformation $u=x-z,$ $v=3y,\text{and}\,w=\frac{z}{2}.$

Evaluate the triple integral $\int_{0}^{2}\,\int_{4}^{6}\,\int_{3z}^{3z+2}(5-4y)dx\,dz\,dy$ by using the transformation $u=x-3z,v=4y,\text{and}\,w=z.$

A transformation $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},T(u,v)=(x,y)$ of the form $x=au+bv,y=cu+dv,$ where $a,b,c,\text{and}\,d$ are real numbers, is called linear. Show that a linear transformation for which $ad-bc\ne 0$ maps parallelograms to parallelograms.

The transformation ${T}_{\theta }:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{\theta }(u,v)=(x,y),$ where $x=u\,\cos \,\theta -v\,\sin \,\theta ,$ $y=u\,\sin \,\theta +v\,\cos \,\theta ,$ is called a rotation of angle $\theta .$ Show that the inverse transformation of ${T}_{\theta }$ satisfies ${T}_{\theta }{}^{-1}={T}_{-\theta },$ where ${T}_{-\theta }$ is the rotation of angle $-\theta .$

[T] Use the results of the previous exercise to find the region $S$ in the $uv\text{-plane}$ whose image through a rotation of angle $\frac{\pi }{4}$ is the region $R$ enclosed by the ellipse ${x}^{2}+4{y}^{2}=1.$ Use a CAS to answer the following questions.

a. Graph the region $S.$ b. Evaluate the integral $\iint_{S}{e}^{-2uv}du\,dv.$ Round your answer to two decimal places.

[T] The transformations ${T}_{i}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},$ $i=1,\ldots ,\,4,$ defined by ${T}_{1}(u,v)=(u,-v),$ ${T}_{2}(u,v)=(-u,v),{T}_{3}(u,v)=(-u,-v),$ and ${T}_{4}(u,v)=(v,u)$ are called reflections about the $x\text{-axis},y\text{-axis},$ origin, and the line $y=x,$ respectively.

a. Find the image of the region $S=\{(u,v)|{u}^{2}+{v}^{2}-2u-4v+1\le 0\}$ in the $xy\text{-plane}$ through the transformation ${T}_{1}\circ {T}_{2}\circ {T}_{3}\circ {T}_{4}.$ b. Use a CAS to graph the image of $S.$ c. Evaluate the integral $\iint_{S}\sin ({u}^{2})du\,dv$ by using a CAS. Round your answer to two decimal places.

[T] The transformation ${T}_{k,1,1}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3},{T}_{k,1,1}(u,v,w)=(x,y,z)$ of the form $x=ku,$ $y=v,z=w,$ where $k\ne 1$ is a positive real number, is called a stretch if $k>1$ and a compression if $0<k<1$ in the $x\text{-direction}.$ Use a CAS to evaluate the integral $\iiint_{S}{e}^{-(4{x}^{2}+9{y}^{2}+25{z}^{2})}dx\,dy\,dz$ on the solid $S=\{(x,y,z)|4{x}^{2}+9{y}^{2}+25{z}^{2}\le 1\}$ by considering the compression ${T}_{2,3,5}(u,v,w)=(x,y,z)$ defined by $x=\frac{u}{2},y=\frac{v}{3},$ and $z=\frac{w}{5}.$ Round your answer to four decimal places.

[T] The transformation ${T}_{a,0}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{a,0}(u,v)=(u+av,v),$ where $a\ne 0$ is a real number, is called a shear in the $x\text{-direction}.$ The transformation, ${T}_{0,b}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{0,b}(u,v)=(u,bu+v),$ where $b\ne 0$ is a real number, is called a shear in the $y\text{-direction}.$

a. Find transformations ${T}_{0,2}\circ {T}_{3,0}.$ b. Find the image $R$ of the trapezoidal region $S$ bounded by $u=0,v=0,v=1,$ and $v=2-u$ through the transformation ${T}_{0,2}\circ {T}_{3,0}.$ c. Use a CAS to graph the image $R$ in the $xy\text{-plane}.$ d. Find the area of the region $R$ by using the area of region $S.$

Use the transformation, $x=au,y=av,z=cw$ and spherical coordinates to show that the volume of a region bounded by the spheroid $\frac{{x}^{2}+{y}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1$ is $\frac{4\pi {a}^{2}c}{3}.$

Find the volume of a football whose shape is a spheroid $\frac{{x}^{2}+{y}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1$ whose length from tip to tip is $11$ inches and circumference at the center is $22$ inches. Round your answer to two decimal places.

[T] Lamé ovals (or superellipses) are plane curves of equations ${\left(\frac{x}{a}\right)}^{n}+{\left(\frac{y}{b}\right)}^{n}=1,$ where a, b, and n are positive real numbers.

a. Use a CAS to graph the regions $R$ bounded by Lamé ovals for $a=1,b=2,n=4$ and $n=6,$ respectively. b. Find the transformations that map the region $R$ bounded by the Lamé oval ${x}^{4}+{y}^{4}=1,$ also called a squircle and graphed in the following figure, into the unit disk. c. Use a CAS to find an approximation of the area $A(R)$ of the region $R$ bounded by ${x}^{4}+{y}^{4}=1.$ Round your answer to two decimal places.

[T] Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation ${\left(\frac{x}{a}\right)}^{n}+{\left(\frac{y}{b}\right)}^{n}=1$ with $\frac{a}{b}=\frac{9}{7}$ and $n=e.$ Use a CAS to find an approximation of the area of the parking garage in the case $a=900$ yards, $b=700$ yards, and $n=2.72$.