Multivariable Calculus
Multivariable calculus extends differentiation and integration to functions of several variables. It introduces partial derivatives, gradients, and multiple integrals.
Key Points
- Partial derivatives measure rate of change along coordinate axes.
- The gradient points in the direction of steepest ascent.
- The multivariable chain rule is essential for backpropagation.
Formulas
Partial derivative
$$\frac{\partial f}{\partial x_i}(x_1, \dots, x_n) = \lim_{h \to 0} \frac{f(x_1, \dots, x_i+h, \dots, x_n) - f(x_1, \dots, x_n)}{h}$$
Gradient
$$\nabla f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n} \right)^\top$$
Multivariable chain rule
$$\frac{\partial f}{\partial t} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \frac{\partial x_i}{\partial t}$$
Code Example
import jax.numpy as jnp
from jax import grad
def f(x):
return jnp.sum(x ** 2)
grad_f = grad(f)
print(grad_f(jnp.array([1.0, 2.0, 3.0]))) # [2, 4, 6]