Gradient

The Gradient

The Gradient of a Real-Valued Functions of 2 or 3 Variables

Consider the function \(f : \mathbb {R}^2\to \mathbb{R}\) defined by \(f(x, y) = (x^2 - y^2)/2\).

The gradient of \(f\) at the point \( (x,y) \) is the vector \(\nabla f(x, y) = \langle x, -y \rangle\).

Here is a DPGraph file showing the graph of \(f\). (See my dpgraph page for information about how to put DPGraph on a Windows machine. Sorry, DPGraph does not work on a Mac, or on most phones.)

This figure shows a contour map of \(f\).
This figure shows \(\nabla f (1,2) \), the gradient of f at the point (1, 2). The gradient vector is perpendicular to the contour, and points uphill.
This figure shows the gradient vector field of \(f\). Note that \(\nabla f: \mathbb {R}^2\to \mathbb{R}^2\). That is, \(\nabla f\) is function whose input is \((x,y) \in \mathbb {R}^2\) and whose output is \(\nabla f(x, y) = \langle x, -y \rangle \in \mathbb{R}^2\). This is our first example of a vector field, which will become a major concept later in the course. We draw the the scaled vectors \( \frac 1 {10} \langle x, -y \rangle \) at grid of (\(x,y)\) points, with their tail at (\(x,y)\).

Now for another function, \(f : \mathbb {R}^2\to \mathbb{R}\) defined by \(f(x, y) = \frac 1 2 x^2 - \frac 1 4 x^4 - \frac 1 2 y^2\).

This figure shows a contour map of \(f\).
This figure shows the gradient vector field, \(\nabla f(x, y) = \langle x - x^3, -y \rangle\).

This DPGraph shows the graph and the gradient vector field of \(f : \mathbb {R}^2\to \mathbb{R}\) defined by \(f(x,y) = 12 + 3 \sin(\pi x/10) \sin(\pi y/10)\). The figure shows how the gradient of \(f\) is a 2-dimensional vector, whereas the graph of \(f\), \( \{ (x,y,z) \in \mathbb{R}^3 \mid z = f(x,y) \} \) is a surface in three dimensions.

The Electric Field is the Negative Gradient of the Electrostatic Scalar Potential

This DPGraph shows the equipotential surfaces, and the electric field for a positive point charge \(Q\) at the origin. The electrostatic potential is \(\varphi: \mathbb {R}^3\to \mathbb{R}\) defined by \[\displaystyle \varphi(x,y,z) = \frac Q {\sqrt{x^2 + y^2 + z^2}} = \frac Q r .\] Following the physics notation, we use \(r = \sqrt{x^2 + y^2 + z^2}\) for the distance from the origin to the point \( (x,y,z)\), rather than \(\rho\) which is the math notation.
The electric field \({\bf E} : \mathbb{R}^3\to \mathbb{R}^3\) is the negative gradient of the electrostatic potential, \( {\bf E}(x,y,z) = - \nabla \varphi(x,y,x)\). A calculation shows that \[ {\bf E}(x,y,z) = \left \langle \frac {Q x} {(x^2+y^2+z^2)^{3/2}}, \frac {Q y} {(x^2+y^2+z^2)^{3/2}}, \frac {Q z} {(x^2+y^2+z^2)^{3/2}} \right \rangle = \frac Q {r^3} \langle x,y,z \rangle .\] The magnitude of the electric field is \( \displaystyle \| {\bf E} (x,y,z) \| = \frac Q {x^2+y^2+z^2} = \frac Q {r^2}\). This is the so-called inverse square law. The electric field points away from the origin at each point.
In the DPGraph, you can use the scrollbar to adjust A, the radius of the sphere, which is an equipotential surface \( \{(x,y,z) \in \mathbb{R}^3 \mid \varphi(x,y,z) = \frac Q A \}\).

This DPGraph shows the equipotential surfaces, and the electric field, for two equal positive point charges placed at (1, 0, 0) and (-1, 0, 0): \[ \varphi(x,y,z) = \frac Q {\sqrt{(x-1)^2 + y^2 + z^2}} + \frac Q {\sqrt{(x+1)^2 + y^2 + z^2}} \] In the DPGraph, the equipotential surface is \( \{ (x,y,z) \in \mathbb{R}^3 \mid \varphi(x, y, z) = A\} \), and you can use the scrollbar to adjust A.

Tangent Planes to Level Surfaces

Consider a function \( f: \mathbb{R}^3 \to \mathbb{R}\). Fix \( (x_0, y_0, z_0) \) and let \(C = f(x_0, y_0, z_0)\). A normal vector to the level surface \( \{ (x,y,z) \in \mathbb{R}^3 \mid f(x,y,z) = C \}\) at the point \( (x_0, y_0, z_0) \) is \( \nabla f(x_0, y_0, z_0)\). So, an equation of the tangent plane to the level surface at the point \( (x_0, y_0, z_0) \) is \[ \nabla f(x_0, y_0, z_0) \cdot \langle x-x_0, y-y_0, z-z_0 \rangle = 0 .\] Here is an ellipsoid with the equation \( (x/6)^2 + (y/5)^2 + (z/4)^2 = 1 \). This ellipsoid is a level surface of the function \(f: \mathbb{R}^3 \to \mathbb{R}\) defined by \( f(x,y,z) = (x/6)^2 + (y/5)^2 + (z/4)^2 \).
Here is the same ellipsoid with the gradient of \(f\), and a plane \(x + y + z = A\). Using the scrollbar, you can adjust A so that the plane is tangent to the ellipsoid. Let \((x_0, y_0, z_0)\) be the point of tangency. Note that \(\nabla f(x_0, y_0,z_0)\) is parallel to \(\langle 1,1,1\rangle\), which is a the normal vector of the tangent plane.

Example: Consider the ellipsoid with the equation \( x^2 + 2 y^2 + 4 z^2 = 22\). Find the points on the ellipsoid where the tangent plane is parallel to the plane with the equation \(2 x + 2 y - 8 z = 0\).
Here is a DPGraph of the ellipsoid and original plane.
Here is a DPGraph of the ellipsoid and the two tangent planes parallel to the original plane.

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