Gradient

The central idea

For differentiable $f:\mathbb R^n\to\mathbb R$, the gradient is $\nabla f=\langle f_{x_1},\ldots,f_{x_n}\rangle\in\mathbb R^n$. It represents the derivative as a vector via $Df(\mathbf a)[\mathbf h]=\nabla f(\mathbf a)\cdot\mathbf h$. At a regular level set, the gradient is normal to tangent directions.

Definitions, hypotheses, and notation

A gradient depends on the chosen inner product and coordinates; in ordinary Cartesian Euclidean space, the standard dot product identifies the derivative covector with this familiar vector. The chain rule along a curve, $d(f\circ\mathbf r)/dt=\nabla f\cdot\mathbf r'$, proves normality to level sets because $f$ is constant along a level curve.

At points where the gradient vanishes, every first-order directional derivative is zero, so second-order or constraint analysis is needed. At a singular level set, normal conclusions may fail because there is no nonzero gradient to determine a unique tangent hyperplane.

Conceptual meaning

The gradient points in the direction of steepest increase, with magnitude equal to the maximum rate per unit distance. Its negative points toward steepest decrease. On contour maps it crosses level curves orthogonally toward larger labels.

A dependable method and decision rule

1. Compute each partial derivative in coordinate order.
2. Evaluate all components at the same point.
3. Treat the result as a vector in the input space.
4. Normalize only when a unit direction is requested.
5. Use dot products for directional rates or normals for level-set tangents.

Fully worked example

Graphical or geometric meaning

Near $(1,2)$, equal-output contours have tangent direction perpendicular to $\langle8,-1\rangle$. Closely spaced contours in the gradient direction reflect the relatively large magnitude $\sqrt{65}$.

Common mistakes and why they fail

Verification and reasonableness checks

• Confirm gradient dimension equals input dimension.
• Test orthogonality against a tangent vector to a level set.
• Compare a small finite change with $\nabla f\cdot\Delta\mathbf x$.

The gradient is normal and points uphill

For differentiable $f$, $\nabla f$ collects partial derivatives and points in the direction of steepest increase. Its magnitude is the maximal rate of change per unit distance. Because $f$ is constant along a smooth level curve or surface, differentiating along a tangent direction gives zero; therefore the gradient is normal wherever it is nonzero. Evaluate at the requested point and normalize only when a unit normal is needed. A zero gradient does not classify the point, and the level set may be singular there. In applications, gradient components carry output units per corresponding input unit; if coordinate units differ, physical interpretation may require scaling rather than treating the coordinate dot product as automatically Euclidean.

Practice

1. Find $\nabla(xy+z^2)$.
2. What is the steepest rate at a point?
3. What does $\nabla f=0$ make the point?

Connections and next steps

• Directional Derivatives
• Tangent Planes
• Lagrange Multipliers