Multivariable Calculus¶
Recall (doc/postdoc). Not a first course.
Job: calculus on \(\mathbb{R}^n\); derivative as linear map; Stokes theorem package.
Core¶
\[
Df_a\in L(\mathbb{R}^n,\mathbb{R}^m),\quad
\nabla f=(\partial_{x_1}f,\ldots,\partial_{x_n}f),\quad
\text{Hessian }H_f=(\partial_i\partial_j f)
\]
| Object | Formula / meaning |
|---|---|
| Directional | \(D_vf=\nabla f\cdot v\) (if \(C^1\)) |
| Chain | \(D(g\circ f)_a=Dg_{f(a)}\circ Df_a\) |
| Jacobian | \(\det Df\) for change of variables |
| Critical | \(\nabla f=0\); classify via \(H_f\) |
| Div / curl | \(\nabla\cdot F\), \(\nabla\times F\) (in \(\mathbb{R}^3\)) |
Integral theorems (oriented, suitable regularity):
\[
\int_a^b F'=F(b)-F(a);\
\oint_{\partial\Omega}P\,dx+Q\,dy=\iint_\Omega(Q_x-P_y);\
\text{Stokes / div thm}
\]
Fubini / Tonelli: product measures; absolute integrability for free reorder.
Used later for¶
- Differential geometry: forms, Stokes on manifolds
- PDE: divergence form, Green's identities
- Optimization: Lagrange multipliers \(\nabla f=\lambda\nabla g\)
- Measure theory: integration on \(\mathbb{R}^n\)
Check¶
If \(\nabla\times F=0\) on \(\mathbb{R}^3\setminus\{0\}\), is \(F\) necessarily a gradient there?
No — e.g. magnetic monopole / \(F=\frac{(-y,x,0)}{x^2+y^2}\) type fields: closed but not exact if domain not simply connected. Need \(H^1_{\mathrm{dR}}(U)=0\).