Calculus
Partial Derivatives
A surface has no single slope, so you have to say which way you are walking. Freeze y and slide a vertical plane through the surface: what is left is one ordinary curve, and the slope of its tangent line is ∂f/∂x. Freeze x instead and the other plane gives you ∂f/∂y.
∂f/∂x—
∂f/∂y—
height f(x, y)—
steepest slope |∇f|—
Drag the scene to orbit, scroll to zoom. Click either slice to select it: the chosen one lights up here and in its equation.
What to observe
- A curve has one slope; a surface has one for every direction you could walk. The two planes pick the two easiest directions: the amber one holdsy still, the blue one holds x still.
- Inside a plane the surface is just a curve, drawn again in the card on the right. That is the whole trick: a partial derivative is an ordinary derivative of that curve, and the other variable is treated as a constant.
- Pick Twist (f = 0.8xy). Slide x and ∂f/∂x never moves, but slide y and it changes sign. The slope along x is allowed to depend on where you are in y: that is why the derivative of a surface needs two numbers, not one.
- On the Saddle, put the point at the centre. Along x the curve bends up, along y it bends down, and both tangent lines are flat at once. Every first derivative vanishes there and the point is still not a minimum.
- Tick the last box. The two tangent lines are two lines through one point, so they span a plane: the tangent plane. Knowing both partials is enough to know how the surface tilts every other way too.
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