Question

Find the gradient vector field for the scalar function. (That is, find the conservative vector field for the potential function.) $$ g(x, y, z)=x y \ln (x+y) $$

Short answer

\(\nabla g = \left(y \ln (x+y) + \frac{x y}{x+y}\right) \hat{i} + \left(x \ln (x+y) + \frac{x y}{x+y}\right) \hat{j}\)

Step-by-step solution

- Understand the gradient operator

The gradient operator is represented as \(\nabla f\), where \(f\) is the function. In three dimensions, it is represented as: \(\nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}\), where \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) are base vectors of the x, y, and z axes, respectively and \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and \(\frac{\partial f}{\partial z}\) are the partial derivative of \(f\) with respect to \(x\), \(y\), and \(z\), respectively.

- Find the partial derivatives

The function \(g(x, y, z) = x y \ln (x+y) \) contains three variables. However, there is no explicit dependence on \(z\), i.e. partial derivative with respect to \(z\) is zero. Let's calculate the partial derivatives with respect to \(x\) and \(y\):\(\frac{\partial f}{\partial x} = y \ln (x+y) + \frac{x y}{x+y}\)and\(\frac{\partial f}{\partial y} = x \ln (x+y) + \frac{x y}{x+y}\)

- Form the Gradient Vector Field

Now we will place our calculated partial derivatives into the gradient vector formula:\(\nabla g = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}\)Substituting our calculated partial derivatives:\(\nabla g = \left(y \ln (x+y) + \frac{x y}{x+y}\right) \hat{i} + \left(x \ln (x+y) + \frac{x y}{x+y}\right) \hat{j} + 0 \hat{k} \)