Gradient Field Discrimination

　　If a vector field F = M i + N j is a gradient field and its potential function is f(x,y), then:

　　So, for a vector field F = M i + N j that is defined everywhere in the plane and is derivable everywhere , if there is M _y = N _x , then this vector field is a gradient field.

Example 1

　　For F = -y i + x j , verify with the above discriminant:

　　So F = -y i + x j is not a gradient field.

Example 2

　　F = (4x ² + axy) i + (3y ² + 4x ² ) j , a is a constant, what is the value of a, F is the gradient field?

find potential function

　　In the above example, F = (4x ² + 8xy) i + (3y ² + 4x ² ) j is the gradient field, so how can I find its potential function? Guessing can be used in simple gradient fields, but this does not always work and a systematic solution must be found.

line integral method

　　The trajectory C of the line integral in the gradient field starts at the origin and ends at (x ₁ , y ₁ ):

　　According to the principle of independent paths, the line integral in the gradient field is only related to the starting point and the end point, and has nothing to do with the path, so a simpler calculation method can be found:

　　According to the fundamental theorem of line integrals:

　　f(0,0) is a constant, so:

　　Now:

　　Remove the subscripts of x and y to get the potential function:

　　Here C is a constant, indicating that f is a family of potential functions, just like adding a constant C to an indefinite integral.

　　verify:

indefinite integral method

　　The potential function f in the gradient field satisfies:

　　According to the Fundamental Theorem of Calculus, the integral of the derivative is equal to the original function, where y is regarded as a constant, and the integral of x is calculated for f _x :

　　Since f _x is a partial derivative, the integral does not add a constant C at the end, but a function g(y) about y.

　　It should be noted that g(y) must not contain x, and only in this way can g(y) be equal to 0 when f _{x is calculated.}

Comprehensive example

Example 1

　　Determine whether a plane vector field is conservative:

　　The premise of a conservative field is that the field is defined everywhere and can be differentiated everywhere. The field here is undefined at x = y = 0, so it can be directly determined that it is not a conservative field.

　　It can also be verified with a little effort that C is the unit circle: