Calculus III: Section 16.3

Professor Ensley

Ship Math

11/18/11

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 1 / 7

Conservative Vector Fields

Conservative Vector Fields

The vector field F is has the path independence property if any integralalong a path from point P to point Q depends only on P and Q, not theparticular path. In this case, we say that F is a conservative vector field.

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 2 / 7

Conservative Vector Fields

Conservative Vector Fields

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 2 / 7

Conservative Vector Fields

Conceptual Insight

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 3 / 7

Conservative Vector Fields

Conceptual Insight

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 3 / 7

Conservative Vector Fields

Equivalent Conditions for Path Independence

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 4 / 7

Conservative Vector Fields

Equivalent Conditions for Path Independence

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 4 / 7

Conservative Vector Fields

Conservative Vector Fields

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 5 / 7

Finding a Potential Function

Finding a Potential Function

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 6 / 7

Finding a Potential Function

Finding a Potential Function

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 6 / 7

Finding a Potential Function

Finding a Potential Function

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 6 / 7

Finding a Potential Function

Finding a Potential Function

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 6 / 7

Finding a Potential Function

Finding a Potential Function

Exercise 12 If the vector field F = 〈z , 1, x〉 is conservative, find apotential function φ(x , y , z).

Exercise 17 If the vector field F = 〈cos z , 2y ,−x sin z〉 is conservative,find a potential function φ(x , y , z).

Exercise 19 If the vector field F = 〈 1x ,−1y 〉 is conservative, find a

potential function φ(x , y).

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 6 / 7

Finding a Potential Function

Conceptual Insight

Example Show that the vortex vector field F =⟨−y

x2+y2 ,x

x2+y2

⟩has equal

cross partials, and then evaluate

∫CF · ds for the path c(t) = 〈cos t, sin t〉

once around the unit circle.

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 7 / 7

Finding a Potential Function

Conceptual Insight

Professor Ensley (Ship Math) Calculus III: Section 16.3 11/18/11 7 / 7