No. 0527 November 2025.
EE2147 vector calculus notes, and Calculus II

Every slope from one vector: the gradient of xy + yz + xz

The gradient gathers a scalar field’s three partial derivatives into one vector. It points the way the field rises fastest, and its length is that fastest rate. The slope in any other direction is the gradient dotted with a unit vector pointing that way. My vector calculus notes work one example: φ = xy + yz + xz at P(1, 2, 3), looking toward Q(3, 4, 4).

∇φ = (y + z) ax + (x + z) ay + (x + y) az

∇φ(1, 2, 3) = 5ax + 4ay + 3az

û = PQ / |PQ| = (2ax + 2ay + az) / 3⇒Dûφ = ∇φ · û = (10 + 8 + 3) / 3 = 7

The steepest slope at P is |∇φ| = √50 ≈ 7.07, so toward Q the field climbs at 99 % of its fastest rate: PQ is only 8.1° off the gradient.

Every slope at P is a chord of one circle. Mark the slope in each direction as a distance from P, and the uphill directions trace a circle whose diameter is the gradient, because D = |∇φ| cos θ. Toward Q the chord is 7; along the level surface it shrinks to nothing. Computed on this page.

what I don’t understand yet

Flux and circulation are in my notes as definitions, ∫A · dS and ∮A · dl. The theorems that tie them to divergence and curl are not, yet. They are the second of four vector-calculus papers I have planned.

where this goes

Entry 10 takes the same vector to photographs: an edge is a large gradient of brightness, found with differences in place of derivatives.