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Calculus II: Vector Calculus and Multivariable Analysis / Vector Calculus and the Integral Theorems

Divergence, Curl, and the Del Operator

The Black-Scholes PDE is Vt+12σ2S2VSS+rSVS−rV=0V_t+\frac{1}{2}\sigma^2 S^2 V_{SS}+rSV_S-rV=0. Rewritten in terms of log-price x=ln⁡Sx=\ln S, this becomes Vt+12σ2Vxx+μVx−rV=0V_t+\frac{1}{2}\sigma^2 V_{xx}+\mu V_x-rV=0 — which contains the Laplacian ΔV=Vxx\Delta V=V_{xx} (in one dimension). The Black-Scholes operator is the drift-modified Laplacian, and the Laplacian is built from the del operator ∇\nabla. This lesson develops divergence ∇⋅F\nabla\cdot\mathbf{F} (net outflow per unit volume), curl ∇×F\nabla\times\mathbf{F} (net rotation per unit area), their algebraic identities, and the Laplacian Δf=∇⋅(∇f)\Delta f=\nabla\cdot(\nabla f) — the differential operator at the heart of diffusion equations, pricing PDEs, and potential theory.
Learning objectives. By the end of this note you should be able to:
  • compute the divergence ∇⋅F\nabla\cdot\mathbf{F} and curl ∇×F\nabla\times\mathbf{F} of a vector field in R3\mathbb{R}^3;
  • state and use the del identities: ∇×(∇φ)=0\nabla\times(\nabla\varphi)=\mathbf{0} and ∇⋅(∇×F)=0\nabla\cdot(\nabla\times\mathbf{F})=0;
  • compute the Laplacian Δf=∇2f\Delta f=\nabla^2 f and identify it as the divergence of the gradient;
  • give the physical interpretation of divergence (source/sink density) and curl (rotation density).

In this lesson

    1. The Del Operator
    1. Divergence
    1. Curl
    1. Del Identities
    1. The Laplacian
    1. Summary of del Operations

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