MA 242.003

• Day 53 – April 2, 2013• Section 13.2: Finish Line Integrals• Begin 13.3: The fundamental theorem for line

integrals

Extension to 3-dimensional space

A major application: Line integral of a vector field along C

A major application: Line integral of a vector field along C

We generalize to a variable force acting on a particle following a curve C in 3-space.

Line Integrals with respect to x, y and z.

Line Integrals with respect to x, y and z.

Line Integrals with respect to x, y and z.

Line Integrals with respect to x, y and z.

Line Integrals with respect to x, y and z.

Line Integrals with respect to x, y and z.

This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.

Line Integrals with respect to x, y and z.

This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.For example, the line integral

Line Integrals with respect to x, y and z.

This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.For example, the line integral

the line integral of the vector field

Result: Any line integral of the form

Result: Any line integral of the form

can be reformulated as a line integral of a vector field

Section 13.3The Fundamental Theorem for Line Integrals

In which we characterize conservative vector fields

Section 13.3The Fundamental Theorem for Line Integrals

In which we characterize conservative vector fields

And generalize the FTC formula

Before we prove this theorem I want to recall a result from the section on the chain rule:

Before we prove this theorem I want to recall a result from the section on the chain rule:

Before we prove this theorem I want to recall a result from the section on the chain rule:

Proof:

(continuation of proof)

See your textbook for the proof!

See your textbook for the proof!

Note that we now have one characterization of conservative vector fields on 3-space.

See your textbook for the proof!

Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path.

See your textbook for the proof!

Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path.

Unfortunately, this characterization is not very practical!

Recall the following theorem from chapter 12:

Recall the following theorem from chapter 12:

Recall the following theorem from chapter 12:

Let’s use this property to investigate the properties of components of a conservative vector field.

Recall the following theorem from chapter 12:

Let’s use this property to investigate the properties of components of a conservative vector field.

(The calculation)

Hence we have proved:

Hence we have proved:

This is another characterization of conservative vector fields!

Hence we have proved:

This is another characterization of conservative vector fields!

The question arises:

Hence we have proved:

This is another characterization of conservative vector fields!

The question arises: Is the CONVERSE true?

Hence we have proved:

This is another characterization of conservative vector fields!

The question arises: Is the CONVERSE true? YES!

Proof given after we study Stokes’ theorem in section 13.7.