Multivariate Calculus – Core Formulas & Concepts

Multivariate Chain Rule

• Total differential for z = f(x,y): dz = fx\,dx + fy\,dy
• Basic chain rule (single parameter t):
  • x = g(t),\; y = h(t)
  • \displaystyle \frac{dz}{dt} = fx\frac{dx}{dt} + fy\frac{dy}{dt}
• Multiple intermediate variables (tree diagram):
  • For w = f(x,y,z) with x(t),\,y(t),\,z(t)
    \displaystyle \frac{dw}{dt} = fx\,x'(t)+fy\,y'(t)+f_z\,z'(t)
  • Paths: multiply derivatives along each branch, add across branches
• Nested dependence example:
  • w = f(x,y),\; x = g(t),\; y = h(u,v),\; u=v(t)
    \displaystyle \frac{dw}{dt}=fx g'(t)+fy\big(hu u'(t)+hv v'(t)\big)

Implicit Differentiation (Shortcuts)

• One dependent variable (2-D): for F(x,y)=k
  \displaystyle \frac{dy}{dx}= -\frac{Fx}{Fy}
• Two dependents (3-D): for F(x,y,z)=k
  \displaystyle \frac{\partial z}{\partial x}= -\frac{Fx}{Fz},\qquad \frac{\partial z}{\partial y}= -\frac{Fy}{Fz}
• Use when explicit solving is hard; compute partials directly on F

Gradient & Directional Derivatives

• Gradient: \nabla f(x,y) = (fx, fy)
• Directional derivative in unit direction \mathbf u:
  \displaystyle D_{\mathbf u} f = \nabla f \cdot \mathbf u
• Properties at a point (a,b):
  • |\nabla f(a,b)| = maximum rate of change
  • Steepest ascent: \mathbf u_{\max}=\frac{\nabla f}{|\nabla f|}
  • Steepest descent: \mathbf u{\min}= -\mathbf u{\max}
  • Zero change: any \mathbf u orthogonal to \nabla f (tangent to level curve)

Riemann Sums in Two Variables

• Partition rectangle [a,b]\times[c,d] into m (x) by n (y) sub-rectangles
• Volume approximation:
  \displaystyle \sum{i=1}^{m}\sum{j=1}^{n} f(x{ij},y{ij})\,\Delta x\,\Delta y
• Common sample points: upper/lower & left/right, or midpoint; choose consistently

Double Integrals & Basic Properties

• Definition as limit of double Riemann sums:
  \iintR f(x,y)\,dA = \lim{m,n\to\infty}\sum{i,j} f(x{ij},y_{ij})\,\Delta x\,\Delta y
• Linearity & comparison (continuous f,g):
  • Additivity over regions
  • \iintR (f+g)=\iintR f+\iint_R g
  • \iintR kf = k\iintR f
  • If f\ge g then \iintR f \ge \iintR g

Fubini’s Theorem (Iterated Integration)

• For continuous f on rectangle [a,b]\times[c,d]:
  \displaystyle \iintR f(x,y)\,dA = \inta^b!\intc^d f(x,y)\,dy\,dx = \intc^d!\int_a^b f(x,y)\,dx\,dy
• Choose the easier order; both give same result