17.1 Vector Fields

(NOT on exam 2)

\( f(x, y) = x + y \) this is also called a scalar field because it assigns a scalar to each point \( (x, y) \).

at \( (1, 1) \) \( f(1, 1) = 2 \)

\( (2, 0) \) \( f(2, 0) = 2 \)

Figure: 2D Cartesian coordinate system with points (1,1) and (2,0) marked on the axes.

an example: temperature distribution in this room

Vector Fields

A vector field is a function that assigns a vector to a point \( (x, y) \).

for example, \( \vec{F}(x, y) = \langle x, y \rangle = x\vec{i} + y\vec{j} \)

at \( (1, 1) \), \( \vec{F}(1, 1) = \langle 1, 1 \rangle = \vec{i} + \vec{j} \)

\( (0, 2) \), \( \vec{F}(0, 2) = \langle 0, 2 \rangle \)

Figure: 2D coordinate system showing vectors <1,1> and <0,2> originating from their respective points.

PAGE 2

the acceleration due to gravity in this room is a vector field

\( \vec{F}(x, y, z) = \langle 0, 0, -g \rangle \)

Figure: 3D coordinate system (x, y, z) with multiple downward-pointing arrows indicating a constant vector field.

magnitude is \( g \)

Example

\( \vec{F}(x, y) = \left\langle \frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right\rangle \)

Sketch?

we can pick points, but let's analyze this vector field a bit more

PAGE 3

\[ \vec{F} = \frac{1}{\sqrt{x^2+y^2}} \langle x, y \rangle \]

\[ |\vec{F}| = \left| \frac{1}{\sqrt{x^2+y^2}} \right| |\langle x, y \rangle| = \frac{1}{\sqrt{x^2+y^2}} \sqrt{x^2+y^2} = 1 \]

So this vector field consists of vectors of magnitude 1.

direction is from \( \langle x, y \rangle \)

at \( (1,1) \rightarrow \langle 1,1 \rangle \)
\( (0,1) \rightarrow \langle 0,1 \rangle \)
\( (1,0) \rightarrow \langle 1,0 \rangle \)
\( (-1,0) \rightarrow \langle -1,0 \rangle \)

Figure: A 2D coordinate plane showing unit vectors pointing radially away from the origin in all directions.

unit vectors pointing radially away from origin

PAGE 4

example

\[ \vec{F}(x,y) = \langle y, -x \rangle \]

magnitude: \( |\vec{F}| = \sqrt{x^2+y^2} \) → the farther from origin, the greater the magnitude

Figure: A vector field plot on a 2D plane where vectors spiral clockwise and increase in length further from the origin.

\( \vec{F}(0,0) = \langle 0,0 \rangle = \vec{0} \) (zero vector)
\( \vec{F}(1,0) = \langle 0,-1 \rangle \)
\( \vec{F}(0,1) = \langle 1,0 \rangle \)
\( \vec{F}(0,-1) = \langle -1,0 \rangle \)
\( \vec{F}(1,1) = \langle 1,-1 \rangle \)
\( \vec{F}(2,0) = \langle 0,-2 \rangle \)

Spiral out

PAGE 5

If the vector field →F is the gradient of some scalar function U, then the function U is called the potential function of the vector field →F.

Example

\[ U = \frac{1}{\sqrt{x^2+y^2}} \]

\[ \vec{\nabla} U = \left\langle \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y} \right\rangle = \left\langle -\frac{1}{2}(x^2+y^2)^{-3/2}(2x), -\frac{1}{2}(x^2+y^2)^{-3/2}(2y) \right\rangle \]

\[ = \left\langle \frac{-x}{(\sqrt{x^2+y^2})^3}, \frac{-y}{(\sqrt{x^2+y^2})^3} \right\rangle \]

So, U is the potential function since the vector field is \( \vec{\nabla} U \).

If we graph them, we see the vectors of the field vector field being \( \perp \) to level curves of U.

PAGE 6

Visualizing Vector Fields and Potential Contours

The following plot illustrates the relationship between a vector field \( \vec{F} \) and the contours of its potential function \( f \). Note how the vectors are consistently perpendicular to the level curves.

Figure: Vector field plot with arrows perpendicular to colored contour lines of a potential function f.

PAGE 7

NOT every vector field is the gradient of something

→not every vector field has a potential function

If the vector field ℒ represents a force vector field, and it has a potential function (ℒ = ∇U), then we say the force is conservative (gravity is one such force)

→in a conservative force field, the work done is independent of path