AcademySeveral-Variable Calculus
Academy
Gradient
Level 1 - Calculus topic page in Several-Variable Calculus.
Principle
The gradient packages all first partial derivatives into a vector. It points in the direction of greatest increase and is perpendicular to level curves or level surfaces.
Notation
\(\nabla f\)
gradient of f
\(f_x,f_y\)
first partial derivatives
\(\mathbf u\)
unit direction vector
The Core Method
For a two-variable function,
Gradient
\[\nabla f=(f_x,f_y)\]
The directional derivative in a unit direction \(\mathbf u\) is
Directional derivative
\[D_{\mathbf u}f=\nabla f\cdot\mathbf u\]
Worked Cases
Question
For \(f(x,y)=x^2+xy\), find \(\nabla f(1,2)\).
Answer
Compute partial derivatives: \(f_x=2x+y\), and \(f_y=x\). Evaluate at \((1,2)\): \(f_x(1,2)=4\), and \(f_y(1,2)=1\). Therefore \(\nabla f(1,2)=(4,1)\).
Examples
Question
If \(\nabla f(0,0)=(3,4)\), find the rate of change in direction \(\mathbf u=(1,0)\).
Answer
Use the dot product: \(D_{\mathbf u}f=(3,4)\cdot(1,0)=3\).