T-8 · Multivariable calculus & vector analysis

Slopes in every direction.

A function of two variables is a landscape. Its gradient points uphill, square to the level curves, and its length is Pythagoras on the two slopes. Green's theorem trades a walk around a boundary for a sum over the inside, and both are computed here so they can be compared. The tangent plane is the best flat fit, and its error is measured.

The gradient at P

∇f = (f_x, f_y) points uphill; its length √(f_x² + f_y²) is the steepest slope, Pythagoras on the two partial slopes.

f(P)	0.35671233
∇f(x, y)	((1 − 2x²)e^(−r²), −2xy·e^(−r²))
∇f(P)	(0.166466, −0.28537)
\|∇f\|	0.33037374
vs finite differences	residual …

Directional derivative

The slope along a unit direction u is ∇f·u = |∇f| cos θ. It is largest along ∇f (Cauchy–Schwarz), zero along the level curve.

u	(0.809, 0.5878) 36°
D_u f = ∇f·u	−0.03306257
by differences	−0.03306257 residual …
\|D_u f\| ≤ \|∇f\|	holds
∇f against the level curve	89.999999° residual …

Hessian and Newton's method

Newton solves ∇f = 0 using the matrix of second derivatives. Near a nondegenerate critical point the error squares at each step, so the digits double.

H(P)	[[−1.627, −0.133], [−0.133, −0.485]] det 0.7714
iterates	5 \|∇f\|: 3.3e−1, 2.0e−1, 2.9e−2, 7.0e−5, 8.9e−10, 1.3e−16
reached	(0.707107, 0) maximum