Method of Lagrange Multipliers

In our seventh class, we began by reviewing Lagrange multipliers and discussing finding extreme values on a surface with an open relation.

Technique of solving the system of equations given by the method of Lagrange multipliers

Let me brief on a general stratage to solve the system of equations given by the method of Lagrange multipliers. We observe that $\lambda$ only appear in the first and the second equation, so if we devide them, the Lagrange multiplier will be eliminate. Put the resulting equation and the third equation (constrain) to gether, we should get a system equation in two equations and two unknows. This system of equation is usually doable. Let me put the above discussion in the following summary. \(\begin{cases} \frac{\partial f}{\partial x} = \lambda\frac{\partial g}{\partial x},\\\\ \frac{\partial f}{\partial y} = \lambda\frac{\partial g}{\partial y},\\\\ g(x,y)=c \end{cases} \Rightarrow \begin{cases} \frac{\partial f/\partial x}{\partial f\partial y} = \frac{\partial g/\partial x}{\partial g/\partial y},\\\\ g(x,y)=c \end{cases}\)

The different between open relation ($g(x,y)\leq c$ or $g(x,y)\geq c$) and close relation $g(x,y)=c$ is that the open relation is a combination of finding extreme values on global and finding extreme values on a constrant.

Use Integral to Find Volume

After a brief history of how we got to where we are today, we introduced double integrals. We started by using single-variable calculus to calculate the length of a graph $y=f(x)$ for $x$ from $a$ to $b$, and then used the disk method to find the volume when we rotated the curve along the $x$-axis (which can be generalized to rotate along any $y=c$), and used the shell method to find the volume of a cylinder given a closed area and rotated along the $y$-axis (generalized version is $x=c$).

Double Integration - Three Levels of Difficulty

Next, we introduced double integrals and discussed what $\Omega$ (Omega) means in the integration $\int\int_\Omega f(x,y)dxdy$. We claimed that the order of $dx$ and $dy$ would not affect the value of the integration in this course, but computationally, sometimes one order of integration might be much easier than the other. These two levels ($dxdy$ and $dydx$ are equivalent easy and hard, and one order is easier than the other order) can only deal with double integrals when $\Omega$ is a rectangle, i.e., $\Omega=[a,b]\times[c,d]$. However, if one side is bounded by a function, then we need the Fubini theorem.

Level 1

We compute the integral by treating the other variable as a constant.

• Example(Assume $\Omega=[a,b]\times[c,d]$)
  • $\int\int_\Omega xydxdy$
  • $\int\int_\Omega e^{x+y}dxdy$
  • $\int\int_\Omega f(x)g(y)dxdy$

Level 2

We do the same like the Level 1, but one order of integration is a dead end.

• Example($\Omega=[a,b]\times[c,d]$)
  • $\int\int_\Omega \frac{y}{\sqrt{x^2+y^2}}dxdy$

Level 3

We will introduce it next time.