Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables.

Partial Derivatives and the Gradient

A partial derivative measures the rate of change of a multivariable function with respect to one variable, holding others constant. The Gradient $\nabla f$ is a vector of partial derivatives: $\nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n} \right)$ It points in the direction of the steepest ascent.

def f(x, y):
  return x**2 + 3*y**2

def gradient_f(x, y):
  # Partial w.r.t x: 2x
  # Partial w.r.t y: 6y
  return (2*x, 6*y)

x0, y0 = 1.0, 2.0
grad = gradient_f(x0, y0)
print(f"Gradient at ({x0}, {y0}): {grad}")

Multiple Integrals

A double integral $\iint_D f(x, y) \, dA$ calculates the volume under the surface $z = f(x, y)$ over a region $D$ in the $xy$-plane.

from scipy.integrate import dblquad

# f(x, y) = x*y
# Over [0, 1] x [0, 2]
area, error = dblquad(lambda y, x: x*y, 0, 1, lambda x: 0, lambda x: 2)

print(f"Double integral of x*y over [0,1]x[0,2]: {area:.4f}")

Vector Fields and Line Integrals

A vector field $\mathbf{F}$ assigns a vector to every point in space. A line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ measures the work done by a force field along a path $C$.

The Jacobian Matrix

The Jacobian matrix of a vector-valued function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ is the matrix of all first-order partial derivatives. It represents the best linear approximation of the function near a point. $J_{ij} = \frac{\partial f_i}{\partial x_j}$