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[2020-02-06 Thu 13:16]
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Session 53: Change of Variables | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

1 Chalkboard

Figure 1: Change variable in double integral

Figure 1: Change variable in double integral

Figure 2: Basic example

Figure 2: Basic example

Figure 3: Basic example continuation

Figure 3: Basic example continuation

Figure 4: Find scaling factor

Figure 4: Find scaling factor

Figure 5: Linear change of variables

Figure 5: Linear change of variables

Figure 6: Example of linear change of variable

Figure 6: Example of linear change of variable

Figure 7: Example: factor scaling

Figure 7: Example: factor scaling

Figure 8: General case

Figure 8: General case

Figure 9: General case 2

Figure 9: General case 2

Figure 10: Jacobian as determinant of partial derivatives

Figure 10: Jacobian as determinant of partial derivatives

2 Why is useful changing the coordinate system in double integrals?

2.1 Front

Why is useful changing the coordinate system in double integrals?

2.2 Back

Because a one better coordinate system is more adapted to the region or integrand than Cartesian coordinate system (x,y)(x,y)

3 What step do we need for change a double integral in polar coordinates?

3.1 Front

What step do we need for change a double integral in polar coordinates?

For example: polar coordinates

\({\displaystyle \iint_R f(x,y) \dd{A}}\)

3.2 Back

  1. Changing the integrand f(x,y)f(x,y) to g(r,θ)g(r,\theta)
    • r=x2+y2r = \sqrt{x^2 + y^2}
    • θ=arctan⁡(y/x)\theta = \arctan(y/x)
    • x=rcos⁡θx = r \cos \theta
    • y=rsin⁡θy = r \sin \theta
  2. Supplying the area element in the r,θr,\theta system: \(\dd{A} = r \dd{r} \dd{\theta}\)
  3. Using the region RR to determine the limits of integration in the r,θr, \theta system.

4 How can we get the area element in another coordinate system?

4.1 Front

How can we get the area element in another coordinate system?

  • u(x,y)u(x,y)
  • v(x,y)v(x,y)

4.2 Back

Using a determinant called the Jacobian

\({\displaystyle \pdv{(x,y)}{(u,v)} = \begin{vmatrix} x_u & x_v \ y_u & y_v \end{vmatrix}}\)

The formula for the area element in the u,vu,v system is

\({\displaystyle \dd{A} = \abs{\pdv{(x,y)}{(u,v)}} \dd{u} \dd{v}}\)

5 What is the formula of change variable for a double integral?

5.1 Front

What is the formula of change variable for a double integral?

\({\displaystyle \iint_R f(x,y) \dd{x} \dd{y}}\)

5.2 Back

\({\displaystyle \iint_R f(x,y) \dd{x}\dd{y} = \iint_R g(u,v) \abs{\pdv{(x,y)}{(u,v)}} \dd{u} \dd{v}}\)

Where g(u,v)g(u,v) is getting from substitution