17.1 Definition and visualization of vector fields

Vector Fields and Scalar Fields

A vector field assigns a vector to every point in some region of space. This gives you a way to model physical quantities that have both magnitude and direction at each location, like fluid velocity, gravitational pull, or electric fields. Scalar fields, by contrast, assign just a number to each point (think temperature or air pressure).

Definition and Components of Vector and Scalar Fields

A vector field in three dimensions is written as a vector-valued function:

$\vec{F}(x, y, z) = \langle P(x, y, z),\; Q(x, y, z),\; R(x, y, z) \rangle$

The functions $P$, $Q$, and $R$ are called component functions. Each one gives the corresponding component of the output vector at a given point. So $P$ controls the x-component, $Q$ the y-component, and $R$ the z-component.

A scalar field is simpler: it's a function $f(x, y, z)$ that outputs a single real number at each point. Temperature distribution across a room is a classic example. You'll often move between scalar and vector fields in this course (for instance, the gradient of a scalar field produces a vector field).

Domain and Range of Vector Fields

• The domain of a vector field is the set of input points where the field is defined. This is typically a subset of $\mathbb{R}^2$ or $\mathbb{R}^3$, depending on whether you're working in two or three dimensions. Physical constraints or the component functions themselves can restrict the domain (e.g., a component with $\frac{1}{r}$ is undefined at the origin).
• The range is the set of all output vectors the field actually produces. It's determined by the component functions evaluated over the domain. Knowing the range helps you understand what magnitudes and directions the field can take on.

2D and 3D Vector Fields

Characteristics and Representation of 2D Vector Fields

A 2D vector field assigns a two-dimensional vector to each point in a region of the plane:

$\vec{F}(x, y) = \langle P(x, y),\; Q(x, y) \rangle$

For example, $\vec{F}(x, y) = \langle -y, x \rangle$ assigns a vector at each point $(x, y)$ that points perpendicular to the position vector. If you sketch a few of these, you'll see a counterclockwise rotation pattern around the origin.

To visualize a 2D field, you draw vector plots (also called arrow plots). Pick a grid of sample points, then draw an arrow at each point whose length and direction match the vector there. The result gives you a qualitative picture of how the field behaves: where it's strong, where it's weak, and how the flow patterns look.

Characteristics and Representation of 3D Vector Fields

A 3D vector field works the same way but with an extra component:

$\vec{F}(x, y, z) = \langle P(x, y, z),\; Q(x, y, z),\; R(x, y, z) \rangle$

Visualizing these is harder because you're placing arrows in three-dimensional space, which gets cluttered fast. Two common strategies:

• Vector plots in 3D: Draw arrows at sample points in space. Software tools handle this better than hand-drawing.
• Cross-sections: Slice the field along a specific plane (say, $z = 0$) and plot the resulting 2D vector field. This lets you analyze behavior one plane at a time while keeping things readable.

Field Lines and Properties

Definition and Characteristics of Field Lines

Field lines (also called streamlines or integral curves) are curves that are tangent to the vector field at every point along them. They trace out the path a particle would follow if it moved in the direction of the field at each instant.

• The density of field lines in a region indicates relative field strength. Where lines are packed closely together, the field is stronger.
• Field lines cannot cross each other. At any given point, the field assigns exactly one vector, so there's only one tangent direction. The one exception is at points where $\vec{F} = \vec{0}$ (called critical points or equilibrium points), where the direction is undefined and field lines may converge or diverge.

Magnitude and Direction Properties of Vector Fields

The magnitude of a vector field at a point tells you the field's strength there. You compute it with the Euclidean norm:

$\|\vec{F}(x, y, z)\| = \sqrt{P^2 + Q^2 + R^2}$

The direction at a point is the orientation of the assigned vector. To isolate just the direction (stripping away magnitude), compute the unit vector:

$\hat{u} = \frac{\vec{F}}{\|\vec{F}\|}$

This is defined everywhere the field is nonzero. Together, magnitude and direction completely characterize the vector field at each point. When you look at a vector plot, arrow length encodes magnitude and arrow orientation encodes direction.