What's Next?
Congratulations
You have completed all 15 lessons of Calculus 3. You can now implement dot and cross products, partial derivatives, gradients, tangent planes, the Laplacian, the second derivative test, and double, polar, and triple integrals — all from scratch in C.
What to Explore Next
- Vector Calculus -- Line integrals, surface integrals, Green's theorem, Stokes' theorem, the divergence theorem
- Differential Equations -- Apply partial derivatives directly to PDEs: the heat equation, wave equation, and Laplace's equation
- Numerical Analysis -- Adaptive quadrature in multiple dimensions, Monte Carlo integration, finite element methods
- Linear Algebra -- The algebra behind gradients and Jacobians: eigenvalues, diagonalization, matrix calculus
Key Formulas
| Concept | Formula |
|---|---|
| Dot product | a·b = ax·bx + ay·by + az·bz |
| Cross product | a×b = (ay·bz-az·by, az·bx-ax·bz, ax·by-ay·bx) |
| Gradient | ∇f = (∂f/∂x, ∂f/∂y) |
| Directional derivative | D_u f = ∇f · û |
| Laplacian | ∇²f = ∂²f/∂x² + ∂²f/∂y² |
| Discriminant | D = fxx·fyy - fxy² |
| Polar area element | dA = r dr dθ |
References
- Calculus by James Stewart — Chapters 12–16 cover multivariable calculus
- 3Blue1Brown: Multivariable Calculus — visual intuition for gradients and partial derivatives
- Numerical Recipes in C — multidimensional integration and optimization