How to Determine if a Vector Field Is Conservative
Calculus III
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Calculus III:
Conservative vector fields
Let's recall that if a vector field 
is conservative then ∫C . 
is independent of path.
This comes from that if the vector field is conservative then it is associated to a potential function f such as ∇f = , and, in turn, using the fondamental theorem of line integrals.
Now, given a vector field , we want to:
• determine whether is conservative, and
• If so, determine its potential function f.
1. Given . Is it conservative ?
Given a vector field, we want to identify if it is conservative.
The vector field is conservative. It is then associated to a potential function f such as ∇f =
= P
+ Q
= (∂f/∂x) + (∂f/∂y). So
∂f/∂x = P
∂f/∂y = Q
Taking the partial derivative, we obtain:
∂2f/∂x∂y = ∂P/∂y
∂2f/∂y∂x = ∂Q/∂x
Therefore
∂P/∂y = ∂Q/∂x
Theorem:
Let = P + Q be a vector field on an open and simply-connected region D.
If P and Q have continuous first order partial derivatives in D and ∂P/∂y = ∂Q/∂x,
then the vector field is conservative.
Example 1
Determine if the vector fields
(x,y) = (x2 + y) + (y2 + x)
is conservative or not.
We have:
P(x,y) = x2 + y
Q(x,y) = y2 + x
∂P/∂y = 1
∂Q/∂x = 1
∂P/∂y = ∂Q/∂x
(x,y) is conservative.
Example 2
Determine if the vector fields
(x,y) = (x2 + xy) + (y2 + xy)
is conservative or not.
We have:
P(x,y) = x2 + xy
Q(x,y) = y2 + xy
∂P/∂y = x
∂Q/∂x = y
∂P/∂y ≠ ∂Q/∂x
(x,y) is not conservative.
2. is conservative.
What is its potential function ƒ ?
Given a conservative vector field, we want to determine the potential function ƒ for this vector field.
Let�s assume that the vector field is conservative and so we know that a potential function, f(x,y) exists. We can then write:
∇f = (∂f/∂x) + (∂f/∂y) = P + Q = .
That is, by setting the two components equal:
∂f/∂x = P, and
∂f/∂y = Q
By integrating each of these, we obtain the two following equations.
f(x,y) = ∫P(x,y) dx or f(x,y) = ∫Q(x,y) dy
Example 3
Determine if the vector field
= (x2y3 + x) + (x3y2 + y)
is conservative and if so, find a potential function for this vector field.
We have:
P(x,y) = x2y3 + x
Q(x,y) = x3y2 + y
∂P/∂y = 3 x2y2
∂Q/∂x = 3 x2y2
∂P/∂y = ∂Q/∂x
So (x,y) is conservative.
Now let�s find the potential function:
f(x,y) = ∫P(x,y) dx = ∫ (x2y3 + x) dx = c(y) is the a function of y.
(1/3)x3y3 + (1/2) x2 + c(y)
This function is constant with respect to x.
f(x,y) = ∫Q(x,y) dy = ∫ (x3y2 + y) dy = c(x) is a function of x.
(1/3)x3y3 + (1/2) y2 + c(x)
This function is constant with respect to y.
Let's work on the first integral:
f(x,y) = (1/3)x3y3 + (1/2) x2 + c(y)
We now need to determine c(y) .
Let�s differentiate f(x,y) with respect to y and set it equal to Q:
∂f/∂y = x3y2 + ∂c(y)/∂y = Q = x3y2 + y.
Hence ∂c(y)/∂y = y , and then
c(y) = (1/2) y2 + const.
Threrfore
f(x,y) = (1/3)x3y3 + (1/2) x2 + (1/2) y2 + const.
The constant const can be anything. So there are an infinite number of possible potential functions, although they will differ by an additive constant.
3. Three-dimentional conservative vector fields
Example 4
Find the potential function for the vector field
(x,y,z) = 2xy2z3 + 2x2yz3 + 3x2y2z2 
We will see in the second next section that the curl of this vector field (x,y,z) in zero. So this vector field is conservative.
Now, we have:
P(x,y,z) = ∂f/∂x = 2xy2z3
Q(x,y,z) = ∂f/∂y = 2x2yz3
R(x,y,z) = ∂f/∂z = 3x2y2z2
Let�s find the potential function:
Let's integrate the first one with respect to x :
f(x,y,z) = x2y2z3 + c(y,z)
• Now, we differentiate this with respect to y and set it equal to Q:
2x2yz3 + ∂c(y,z)/∂y = 2x2yz3
Hence ∂c(y,z)/∂y = 0. That is c(y,z) = c(z)
Therefore
f(x,y,z) = x2y2z3 + c(z)
• Now, we differentiate with respect to z and set the result equal to R:
3x2y2z2 + ∂c(z)/∂z = 3x2y2z2
Hence ∂c(z)/∂z = 0. That is c(z) = const.
The potential function for this vector field is then:
f(x,y,z) = x2y2z3 + const.
How to Determine if a Vector Field Is Conservative
Source: https://scientificsentence.net/Equations/CalculusIII/index.php?key=yes&Integer=conservative_vector_fields