Conservative Vector Fields
Introduction
If the vector field is conservative, a line integral can be solved using the fundamental theorem of line integrals.
The Essentials
For the vector function \( \vec{F}=\langle P,Q,R\rangle \), then \( P_y=Q_x \), \( P_z=R_x \), and \( Q_z=R_y \) if and only if \( \vec{F} \) is conservative. If the vector function is conservative, then a potential function can be found and the fundamental theorem of line integrals can be used to calculate the line integral. The fundamental theorem of line integrals is:
Where \( \nabla f = \vec{F} \). The potential function \( f \) can be found from a two dimensional \( \vec{F} \) using this method if it is conservative: First, integrate \( P \) with respect to \( x \). The \( y \) will be treated as a constant, so the constant of integration will be \( h(y) \) instead of \( C \). Second, take the partial derivative of the result and set it equal to \( Q \) to find what \( h'(y) \) is. Third, integrate \( h'(y) \) to find \( h(y) \) and plug it back in to find the potential function. This method can be generalized to work in three dimensions.
Example
Use the fundamental theorem of line integrals to calculate the line integral of the function from the point (0,0) to the point (1,1):
Where \( \vec{F}=\langle 3x^2y^2e^y+2xy,2x^3ye^y+x^3y^2e^y+x^2\rangle \) First, we test to see if the function is conservative. Since this problem has just two variables, we only have to test \( P_y=Q_x \):
They are equal which means the vector function is conservative so we can find a potential function. Since \( f_x=P \), we will start to find the potential function by taking \( P \) and integrating with respect to \( x \):
Since y was constant in the integration, the constant of integration is a function of \( y \), shown by \( h(y) \) in the equation. Since \( f_y=Q \), we can take the partial derivative of \( f \) with respect to \( y \) and set it equal to \( Q \) in order to find \( h'(y) \):
So \( h'(y) = 0 \) and \( h(y)=C \). Therefore the potential function is:
Now to use the fundamental theorem:
Practice
Verify that the function is conservative, then use the fundamental theorem of line integrals to solve the line integral of \( \vec{F} \):
From the point (0,2) to the point (3,5).
Solution:
0