Question

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. $$ \mathbf{F}(x, y)=3 x^{2} y^{2} \mathbf{i}+2 x^{3} y \mathbf{j} $$

Short answer

If the curl of the vector field \( \mathbf{F}(x, y)=3 x^{2} y^{2} \mathbf{i}+2 x^{3} y \mathbf{j}\) is equal to zero, this implies that the field is conservative. The potential function can then be determined by integrating each component of the field with respect to their respective variables.

Step-by-step solution

Compute partial derivates

Let's first compute the partial derivatives of the given vector field \(\mathbf{F}(x, y)=3 x^{2} y^{2} \mathbf{i}+2 x^{3} y \mathbf{j}\). The first component \(M(x, y)\) is \(3 x^{2} y^{2}\) and the second component \(N(x, y)\) is \(2 x^{3} y\). Now, find \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\). Using these derivatives, it's possible to check the curl of the field.

Check if curl equals to zero

Now, compute the curl of the vector field, which is \(\nabla × F = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\). If this difference equals to zero, then the given vector field is conservative.

Calculate the potential function

Given that the field is conservative, let's move to calculating the potential function. To do this, integrate \(M(x, y)\) with respect to \(x\) and \(N(x, y)\) with respect to \(y\). The potential function \(f(x, y)\) can be found from these integrations.

Key concepts

Potential Function

In the realm of vector calculus, a potential function is a scalar function whose gradient is equal to a given vector field in certain conditions. When identifying a potential function for a conservative vector field, it implies there exists a function, let's denote it as \( f(x, y) \), such that \( abla f = \mathbf{F} \), where \(abla\) represents the gradient operator and \(abla f\) yields the original vector field \( \mathbf{F} \).

For the exercise given, we are provided with the vector field \( \mathbf{F}(x, y) =3 x^{2} y^{2} \mathbf{i}+2 x^{3} y \mathbf{j} \). Identifying if this field is conservative is pivotal, as only a conservative field guarantees a single-valued potential function. Assuming it is conservative, finding the potential function involves integrating the components of the field with respect to their variables while considering any constant of integration as a function of the other variable. If successful, the derived function will encapsulate the essence of the field, providing a powerful tool for further analysis.

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They measure how a function changes as each variable is varied while keeping other variables constant. Represented as \( \frac{\partial}{\partial x} f(x,y) \) for the derivative with respect to \( x \), and similarly for \( y \), these derivatives provide insight into the function's behavior in each dimension separately.

When the vector field \( \mathbf{F} \) from our example is broken down into its components \( M(x,y) = 3x^2y^2 \) and \( N(x,y) = 2x^3y \), the task usually involves computing \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \). These operations yield critical information for determining whether the field is conservative by showing whether the mixed partial derivatives are equal, leading to a curl of zero. This step is indispensable as it lays down the groundwork for further steps involving the potential function if the field is indeed conservative.

Curl of a Vector Field

The curl of a vector field is a concept that measures the rotation or twisting effect the field exhibits at a point. In three dimensions, it is another vector field, but when looking at two-dimensional vector fields like in the exercise, the curl is a scalar that represents the difference between the partial derivatives of the field's components.

For our given two-dimensional vector field \( \mathbf{F} \), the curl is computed by the formula \(abla × \( \mathbf{F} \) = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\). If this difference is zero throughout the region of interest, then the field is said to be irrotational, which is another way of describing a conservative vector field. In our step-by-step solution, the verification that the curl equals zero is crucial; it's the checkpoint that confirms the vector field's candidacy for having a potential function. Understanding the curl and its implications on the nature of the vector field is an essential skill in vector calculus.