Determining Path Independence and Potential Function for a Vector Field
In the realm of multivariable calculus, the concept of vector fields is a crucial topic. A vector field is a function that assigns a vector to each point in a given region of space. One important property of a vector field is its path independence, also known as being a conservative field.
In this article, we will examine the vector field F(x,y) = (x+5) i + (6y+5) j and determine whether it is path independent (conservative) or not. If it is path independent, we will also find a potential function for the vector field.
Defining the Vector Field
The given vector field is:
F(x,y) = (x+5) i + (6y+5) j
where i and j are the standard unit vectors in the x and y directions, respectively.
Determining Path Independence
A vector field F(x,y) = P(x,y) i + Q(x,y) j is said to be path independent (conservative) if the following condition is satisfied:
∂P/∂y = ∂Q/∂x
In other words, the partial derivatives of the component functions P(x,y) and Q(x,y) with respect to the opposite variables must be equal.
Let's calculate the partial derivatives for the given vector field:
∂P/∂y = 0
∂Q/∂x = 6
Since ∂P/∂y ≠ ∂Q/∂x, the vector field F(x,y) = (x+5) i + (6y+5) j is not path independent (conservative).
Finding a Potential Function (Optional)
If a vector field is path independent (conservative), it means that there exists a scalar function U(x,y), called the potential function, such that:
F(x,y) = ∇U(x,y)
where ∇ represents the gradient operator.
Since the given vector field is not path independent, there is no potential function that can be found for it.
Conclusion
In this article, we have analyzed the vector field F(x,y) = (x+5) i + (6y+5) j and determined that it is not path independent (conservative). This means that there is no potential function that can be found for this vector field.
Understanding the concept of path independence and the ability to determine whether a vector field is conservative or not is an essential skill in multivariable calculus. This knowledge is crucial for various applications, such as in physics, engineering, and other scientific fields.