06 Multiple Integrals in Curvilinear Coordinates Free Online FlashCards

What does a Jacobian measure?

The Jacobian is the local area-scaling factor of a two-dimensional transformation, or the local volume-scaling factor in three dimensions.

What are the three steps in a coordinate change?

Rewrite the integrand, transform the region and bounds, and replace the area or volume element with the absolute Jacobian.

What are the Cartesian-to-polar formulas?

The polar conversion formulas are $x=r\cos\theta$ and $y=r\sin\theta$, where $r$ is distance from the origin.

What is the polar area element?

The planar area element is $dA=r\,dr\,d\theta$. The factor $r$ comes from the polar Jacobian.

What are polar bounds for $x^2+y^2\le a^2$?

Use $0\le r\le a$ and $0\le\theta\le2\pi$. These bounds cover the entire disk exactly once.

How do cylindrical coordinates extend polar coordinates?

Cylindrical coordinates use $x=r\cos\theta$, $y=r\sin\theta$, and leave $z$ unchanged.

What is the cylindrical volume element?

The cylindrical volume element is $dV=r\,dr\,d\theta\,dz$. Its Jacobian is $r$.

How are spherical angles defined?

In the standard convention, $\theta$ is the azimuthal angle in the $xy$-plane, while $\phi$ is measured downward from the positive $z$-axis.

What are the Cartesian-to-spherical formulas?

The spherical conversion is $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, and $z=\rho\cos\phi$.

What is the spherical volume element?

The spherical volume element is $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$.

Which spherical angle describes $z=\sqrt{x^2+y^2}$?

The cone $z=\sqrt{x^2+y^2}$ becomes $\phi=\frac{\pi}{4}$.

What are cylindrical bounds for the upper hemisphere?

For the upper hemisphere, use $0\le r\le R$, $0\le\theta\le2\pi$, and $0\le z\le\sqrt{R^2-r^2}$.