MA 242.003 Day 62 – April 15, 2013 Section 13.6: Surface integrals

How do we evaluate such an integral?

How do we evaluate such an integral? Recall our approximation of surface area:

The surface integral over S is the “double integral of the function over the domain D of the parameters u and v”.

This formula should be compared to the line integral formula

The surface integral of f(x,y,z) = 1 over S yields the Notice the special case: The surface integral of f(x,y,z) = 1 over S yields the “surface area of S”

This is the problem we didn’t finish in the last lecture.

(continuation of example)

(continuation of example)

(continuation of example)

Surface integrals of vector fields Goal: To compute how much a vector field “cuts through a surface S”.

The vector field is everywhere normal to the surface

The vector field is NOT normal to the surface

Only the component of F parallel to the unit normal vector n crosses the surface S

Only the component of F parallel to the unit normal vector n crosses the surface S So we want to INTEGRATE the normal component of F over the surface S.

Question: Which direction is the POSITIVE direction across the surface S?

Answer: Either direction can be chosen as the positive direction. Question: Which direction is the POSITIVE direction across the surface S? Answer: Either direction can be chosen as the positive direction.

Is a choice of ORIENTATION for the surface. Terminology: Selecting one of the two choices for the positive direction Is a choice of ORIENTATION for the surface.

Is a choice of ORIENTATION for the surface. Terminology: Selecting one of the two choices for the positive direction Is a choice of ORIENTATION for the surface.

Is a choice of ORIENTATION for the surface. Terminology: Selecting one of the two choices for the positive direction Is a choice of ORIENTATION for the surface.

Question: Are all surfaces in 3-space orientable?

Question: Are all surfaces in 3-space orientable? Answer: No!! The classic example is the Mobius Strip.

Question: Are all surfaces in 3-space orientable? Answer: No!! The classic example is the Mobius Strip.

Question: Are all surfaces in 3-space orientable? Answer: No!! The classic example is the Mobius Strip.

Question: Are all surfaces in 3-space orientable? Answer: No!! The classic example is the Mobius Strip. Orientable surfaces have TWO SIDES The Mobius strip has only ONE SIDE

Question: Are all surfaces in 3-space orientable? Answer: No!! The classic example is the Mobius Strip. Orientable surfaces have TWO SIDES The Mobius strip has only ONE SIDE Here is a YouTube video showing that a Mobius strip Has only one side.

You should be aware that not all surfaces are orientable, You should be aware that not all surfaces are orientable,. However all the surfaces in the problems we work will be orientable.

Recall that only the component of F parallel to the unit normal vector n crosses the surface S

Recall that only the component of F parallel to the unit normal vector n crosses the surface S So we want to INTEGRATE the normal component of F over the surface S. F

Recall that only the component of F parallel to the unit normal vector n crosses the surface S So we want to INTEGRATE the normal component of F over the surface S. F

How are orientations stated in problems?

How are orientations stated in problems? Examples:

This is simply a special surface integral, so we evaluate it as follows: