Skip to content

Multi-variable Calculus

Notation

\[ \nabla \equiv \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \]

Gradient

\[ \operatorname{grad} \phi = \nabla \phi \]
  • computed for a scalar valued function and returns a vector valued function.
  • it point in the direction fof steepest change

Directional Derivative

Divergence

\[ \operatorname{div} \textbf{A} = \nabla \cdot \textbf{A} \]

Curl

\[ \operatorname{curl} \textbf{A} = \nabla \times \textbf{A} \]

Change of Variables

If,

  • \(z = f(x,y)\)
  • \(x = \phi(t)\)
  • \(y = \phi(t)\)

Then,

\[ \frac{df}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{ty} \]

If,

  • \(z = f(x,y)\)
  • \(x = \phi(u, v)\)
  • \(y = \phi(u, v)\)

Then,

\[ \frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial u} \]
\[ \frac{\partial f}{\partial v} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial v} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial v} \]