Multivariable Graphical Techniques in Calculus

Scalar and Vector Fields

• In multivariable calculus, two kinds of fields exist: scalar and vector. A scalar field is a pure numeric construct, having no sense of direction or movement. For example, consider a landscape rendered into a three-dimensional map of magnitudes, where the numeric values represent elevation levels at any given point. It's descriptive of a static circumstance.
  A vector field is composed of vectors instead of points, so it has both magnitude and direction. For example, consider a graph of the magnetic fields around the earth. These fields are never static. Arrows are drawn emerging from the magnetic North Pole, circling the globe and entering the magnetic South Pole. From scalar or vector fields come three important operators: gradient, divergence and curl.

Gradient

• The gradient is a vector field applied to a scalar field. It determines the directions in which magnitudes are changing. For example, taking the gradient of the data responsible for constructing a hilly landscape's topographical map results in a vector field, which can be thought of as lying atop the original field. This gradient field is composed of arrows, which point the way from valleys to individual hilltops.

Divergence

• Divergence applies to vector fields, expressing the magnitude of source or sink points across the vector field. Divergence ultimately overlays a vector field with an assignment of positive or negative scalar measurements. For example, consider the magnetic field vector field. The divergence operator will show major sources or sinks at the magnetic poles and also reveal areas across the globe where minor sinks and sources are found.

Curl

• Curl can be applied to a three-dimensional vector field; it measures infinitesimal rotations in that field. For example, consider a vector field equating to the flow of water through the drain of a kitchen sink. The graphical representation of this motion wouldn't be a simple straight line through the drain, since water rotates like a funnel around the drain itself. Curl would express this rotation in the form of a separate vector field.