6.2 The gradient vector and its geometric interpretation

Gradient Vector and Nabla Operator

The gradient vector captures two essential pieces of information about a scalar function: the direction in which the function increases most rapidly, and the rate of that increase. Understanding the gradient geometrically connects the algebra of partial derivatives to the shape and behavior of surfaces in higher dimensions.

Definition and Notation

The gradient of a scalar function $f$ is denoted $\nabla f$ (read "del f" or "nabla f"). The symbol $\nabla$ is the nabla operator, a vector differential operator defined in Cartesian coordinates as:

$\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$

When you apply $\nabla$ to a scalar function $f(x, y, z)$, you get the gradient vector:

$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$

Each component is just the partial derivative of $f$ with respect to that coordinate. For a function of two variables $f(x,y)$, the gradient lives in $\mathbb{R}^2$ and has only the $x$ and $y$ components.

Scalar and Vector Fields

A scalar field assigns a single number to every point in space. Temperature at each point in a room or the elevation of a terrain are scalar fields. A vector field assigns a vector to every point, like the velocity of a fluid at each location.

The gradient bridges these two ideas: it takes a scalar field as input and produces a vector field as output. At every point in the domain, $\nabla f$ gives you a vector that encodes both the direction and magnitude of the steepest increase in $f$.

Gradient Computation

To compute $\nabla f$, take the partial derivative with respect to each variable and assemble the results into a vector.

Example: Let $f(x, y, z) = x^2 y + e^z$.

1. $\frac{\partial f}{\partial x} = 2xy$
2. $\frac{\partial f}{\partial y} = x^2$
3. $\frac{\partial f}{\partial z} = e^z$

So $\nabla f = (2xy,\; x^2,\; e^z)$.

At the point $(1, 3, 0)$, this evaluates to $\nabla f = (6, 1, 1)$. The function increases fastest in the direction of $(6, 1, 1)$, and the rate of that maximum increase is $|\nabla f| = \sqrt{36 + 1 + 1} = \sqrt{38}$.

Geometric Interpretation

Level Curves and Surfaces

A level curve (in 2D) or level surface (in 3D) is the set of all points where $f$ takes a constant value $c$. Think of contour lines on a topographic map: each line connects points at the same elevation.

The gradient vector at a point is always perpendicular to the level curve or level surface passing through that point. This is one of the most important geometric facts about the gradient. It means:

• $\nabla f$ points "across" level curves, from lower values toward higher values.
• The magnitude $|\nabla f|$ tells you how tightly packed the level curves are. Where the gradient is large, level curves are close together (the function is changing rapidly). Where the gradient is small, level curves are spread apart.

If you're standing on a hillside, the gradient points straight uphill, perpendicular to the contour line under your feet.

Normal Vector and Tangent Plane

For a level surface defined by $f(x, y, z) = c$, the gradient $\nabla f$ evaluated at a point $\mathbf{p}$ on that surface serves as a normal vector to the surface at $\mathbf{p}$.

This gives you a direct way to write the equation of the tangent plane at $\mathbf{p} = (x_0, y_0, z_0)$:

$f_x(x_0,y_0,z_0)(x - x_0) + f_y(x_0,y_0,z_0)(y - y_0) + f_z(x_0,y_0,z_0)(z - z_0) = 0$

This works because the tangent plane consists of all displacement vectors from $\mathbf{p}$ that are perpendicular to $\nabla f$, and "perpendicular" means the dot product with $\nabla f$ is zero.

Orthogonality and Gradient Magnitude

Two vectors are orthogonal when their dot product equals zero. Any vector tangent to a level surface at a point must be orthogonal to $\nabla f$ at that point. This is why the gradient is normal to the surface: it's perpendicular to every possible tangent direction along the surface.

The magnitude $|\nabla f|$ has a direct interpretation as the maximum rate of change of $f$ per unit distance. A few key cases:

• Large $|\nabla f|$: the function value is changing rapidly. Level curves/surfaces are closely spaced.
• Small $|\nabla f|$: the function is nearly flat in that region.
• $|\nabla f| = 0$: this is a critical point. The function has no preferred direction of increase, which occurs at local maxima, local minima, or saddle points. No tangent plane is defined by the gradient at such points (since the normal vector is the zero vector).