What is gradient? If \(\phi=3(x^2y-y^2x)\), find \(\nabla\phi\) at the point (1, -2, -1).

Gradient: The gradient of a scalar field \(\phi(x,y,z)\) is the vector

\[ \nabla\phi=\frac{\partial\phi}{\partial x}\hat i+\frac{\partial\phi}{\partial y}\hat j+\frac{\partial\phi}{\partial z}\hat k \]

Its magnitude is the maximum rate of change of \(\phi\) with distance, and it points in the direction of that maximum increase, normal to the level surface \(\phi=\) constant. Example: \(\vec E=-\nabla V\).

Given: \(\phi=3x^2y-3xy^2\)

\(\dfrac{\partial\phi}{\partial x}=6xy-3y^2\), \(\dfrac{\partial\phi}{\partial y}=3x^2-6xy\), \(\dfrac{\partial\phi}{\partial z}=0\)

At (1, −2, −1):

• \(6(1)(-2)-3(4)=-12-12=-24\)
• \(3(1)-6(1)(-2)=3+12=15\)

\(\nabla\phi=-24\,\hat i+15\,\hat j\)

Magnitude: \(|\nabla\phi|=\sqrt{576+225}=\sqrt{801}\approx28.3\)