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