Change of Variables - Find Integral Bounds

I am having trouble with determining the integral bounds in change of variable problems.

The problem:

Consider the transformation given by $$\space x = u - \sqrt{(\frac13)}\cdot v \space$$ and $$\space y = u + \sqrt{(\frac13)}\cdot v$$ $$- g(u,v) \space$$ and $$h(u,v)$$ respectively

$$R = {(x,y)\in R^2 : x^2 −xy+y^2 \le1}$$ and $$S = {(u,v)∈ R^2 : (u - \sqrt{(\frac13)}\cdot v, u + \sqrt{(\frac13)\cdot v)} \in R}$$

I understand that we take the integral of $$f(g(u,v), h(u,v))\cdot\text{Jacobian} - \text{Jacobian}$$ evaluates to $$\frac{2}{\sqrt{(\frac13)}}$$

This is where I get stuck. I know we can find the bounds for $$x$$ and $$y$$ from $$x^2 −xy+y^2 \le1$$ but I'm unsure how to convert that to $$u,v$$ variables.

In the problem I have to show this $$$$\iint_Rx^2-xy+y^2\ dA=\frac{\pi}{\sqrt{3}}$$$$

And also have to draw the transformation from $$(x,y)$$ to $$(u,v)$$

• My question is what is your region R? What is your area bounded by? Commented Jun 18, 2020 at 19:52
• The $x^2-xy+y^2$ is the multivariable function you are integrating. Commented Jun 18, 2020 at 19:54
• Is the problem Type I or Type 2, is it $dydx$, or $dxdy$ Commented Jun 18, 2020 at 19:55
• To me at least your problem is missing the initial region, whether it's type 1 or type 2, the point of the transformation, your determinant on how you got your Jacobian, your partials. Things that would help solve the problem. Commented Jun 18, 2020 at 20:03
• if $x=u-\frac{1}{\sqrt{3}}v$, and $y=u+\frac{1}{\sqrt{3}}v$, then you Jacobian is setup as follows. \begin{bmatrix}\frac{\delta x}{\delta u}&\frac{\delta x}{\delta y}\\ \frac{\delta y}{\delta u} & \frac{\delta y}{\delta v}\end{bmatrix} Commented Jun 18, 2020 at 20:25

1 Answer

Consider the assignment of the variables $$u(x, y) = \dfrac{x + y} 2$$ and $$v(x, y) = \dfrac{\sqrt 3(y - x)} 2.$$ We have that $$u^2 + v^2 = \frac{x^2 + 2xy + y^2} 4 + \frac{3(y^2 - 2xy + x^2)} 4 = x^2 - xy + y^2.$$ Consequently, we obtain a transformation $$G(x, y) = (u(x, y), v(x, y))$$ with Jacobian $$\operatorname{Jac}(G) = \det \begin{pmatrix} \frac 1 2 & -\frac{\sqrt 3} 2 \\ \frac 1 2 & \phantom -\frac{\sqrt 3} 2 \end{pmatrix} = \frac{\sqrt 3}{2}.$$ Observe that the region in the $$uv$$-plane that is mapped onto by $$G$$ is given by $$0 \leq u^2 + v^2 \leq 1,$$ i.e., the disk of radius $$1$$ centered at the origin. Using polar coordinates, this region is $$\{(r, \theta) \,|\, 0 \leq r \leq 1 \text{ and } 0 \leq \theta \leq 2 \pi\}.$$ By the Change of Variables Formula, therefore, we have that $$\iint_R (x^2 - xy + y^2) \, dA = \frac 2 {\sqrt 3} \iint_U (u^2 + v^2) \, dA = \frac 2 {\sqrt 3} \int_0^{2 \pi} \int_0^1 r^3 \, dr \, d \theta = \frac{\pi}{\sqrt 3}.$$