91Ó°ÊÓ

Partial derivatives help us understand how a function changes as we change one of its variables while keeping the other variables constant.
In the context of our exercise, we are working with a vector field \( \mathbf{F}(x, y) = P\mathbf{i} + Q\mathbf{j} \). To determine conservatism, we focus on two functions here: \( P(x, y) = \ln y + 2xy^3 \) and \( Q(x, y) = 3x^2y^2 + \frac{x}{y} \).
We need to compute the partial derivatives \( \frac{\partial P}{\partial y} \) and \( \frac{\partial Q}{\partial x} \) and check if they are equal.

• \( \frac{\partial P}{\partial y} = \frac{1}{y} + 6xy^2 \)
• \( \frac{\partial Q}{\partial x} = 6xy^2 + \frac{1}{y} \)

By verifying these are equal, we confirm \( \mathbf{F} \) is conservative, as their equality ensures potential alignment in all dimensions.

A potential function \( f(x, y) \) serves as a kind of 'landscape' representation of the vector field.
If a field is conservative, we can find this potential function such that the gradient of \( f \) matches our vector field \( \mathbf{F} \).
In this exercise, finding \( f \) involves integrating the function \( P(x, y) \) with respect to \( x \).

• \( f(x, y) = \int (\ln y + 2xy^3) \, dx = x\ln y + x^2y^3 + g(y) \)

Here, \( g(y) \) is an unknown function solely dependent on \( y \), and it reflects the inherent freedom in integration constants when working in multiple dimensions.
After integrating and realizing that \( Q(x, y) \) provides a check for \( \frac{\partial f}{\partial y} \), we ensure the full alignment of the given field with our derived potential function.

The gradient of a function, denoted as \( abla f \), is a vector that points in the direction of the greatest rate of increase of the function.
It can be considered as a generalization of a derivative to multiple dimensions.
When calculating \( abla f = \mathbf{F} \), and successfully matching these two, we confirm the conservative nature of the field.

• The components of \( abla f \) are the partial derivatives of \( f(x, y) \) with respect to each variable.
• Achieving \( abla f = \mathbf{F} \) validates that our function \( f \) accurately describes the potential landscape of \( \mathbf{F} \).

This reflection ensures that our journey to find \( f(x, y) \) has been executed correctly, affirming the step-by-step integrity laid out in the solution process, from partial derivatives to finally aligning the gradient.