Scalar and vector fields, level surfaces

Scalar fields. A scalar field is a function that takes a point of space and returns a single real number:

where: $(x,y,z)$ is a point in the domain region and $w$ is the single real number (temperature, pressure, density, potential, height) that $f$ assigns to it. In two dimensions the same idea reads $w=f(x,y)$, useful for quantities defined over a flat surface such as a metal plate or a map.

Vector fields. A vector field takes a point of space and returns a vector — a magnitude and a direction attached to that point:

where: $F_1, F_2, F_3$ are themselves scalar fields — the components of the vector along $\hat\imath,\hat\jmath,\hat k$ at each point. Fluid velocity, electric and gravitational force, and wind over a region are the standard physical examples; each assigns a full vector, not just a number, to every point.

Level surfaces and level curves. Given a scalar field $f(x,y,z)$, its level surface at height $c$ is the entire set of points that share that one output value:

where: $c$ is a fixed real constant chosen from the range of $f$, and the level surface is the locus of all points $(x,y,z)$ satisfying the equation — generally a two-dimensional surface sitting inside three-dimensional space. In two variables the analogous object is a level curve, $f(x,y)=c$, a one-dimensional curve in the plane — exactly the isotherms on a weather map or the contour lines on a hiking map. Two structural facts follow directly from $f$ being a function (one output per input): level surfaces for different constants never intersect, and they sit closer together where $f$ is changing rapidly and farther apart where $f$ is nearly constant — a spacing cue that becomes precise once the gradient vector is introduced in the next topic.

Problem. A metal plate occupies the $xy$-plane and its temperature is the scalar field $T(x,y)=x^2+y^2$ (temperature in degrees Celsius, $x,y$ in metres). Describe the level curves of $T$ and sketch the ones for $T=1,4,9$.

Step 1. Set up the level-curve equation. A level curve is $T(x,y)=c$:

Step 2. Interpret the geometry. For $c>0$ this is a circle of radius $\sqrt c$ centred at the origin; for $c=0$ it collapses to the single point $(0,0)$; for $c<0$ there is no curve, since $x^2+y^2\ge 0$ always, matching the fact that $T$ (a sum of squares) can never be negative.

Step 3. Plug in the three requested values. $T=1$ gives $x^2+y^2=1$, a circle of radius $1$ m; $T=4$ gives $x^2+y^2=4$, radius $2$ m; $T=9$ gives $x^2+y^2=9$, radius $3$ m.

Step 4. Read the physics. The plate is coolest at the origin, where $T=0$, and heats up radially outward along concentric circular isotherms. The radii $1,2,3$ m are equally spaced while the temperatures $1,4,9$ are not, so isotherms drawn at equal temperature increments — say $T=1,2,3,4,\dots$ — would crowd together near the centre and spread apart farther out, showing where $T$ changes fastest.

The level curves of $T(x,y)=x^2+y^2$ are concentric circles of radius $\sqrt c$, giving radii $1$ m, $2$ m, and $3$ m for $T=1,4,9$.

Competitive stretch

Problem. Classify $g(x,y,z)=2x-y+3z$ as a scalar or vector field, and if scalar, describe its level surfaces and the spacing between the surfaces $g=0$ and $g=1$.

Step 1. Classify. $g$ returns one real number per point $(x,y,z)$, so it is a scalar field.

Step 2. Write the level-surface equation. Setting $g(x,y,z)=c$ gives

a linear equation in $x,y,z$, so every level surface is a plane, with normal direction $(2,-1,3)$ fixed for all $c$ — the family is a stack of parallel planes.

Step 3. Find the spacing. The perpendicular distance between the planes $g=0$ and $g=1$ is $\dfrac{|1-0|}{\sqrt{2^2+(-1)^2+3^2}}=\dfrac{1}{\sqrt{14}}$ metres, using the standard point-to-plane distance formula with the coefficient vector as normal — unlike the spherical family above, a linear field's level surfaces are evenly spaced, since $g$ changes at the same rate everywhere.

The level surfaces of $g$ are parallel planes normal to $(2,-1,3)$, spaced $1/\sqrt{14}\ \text{m}$ apart per unit change in $g$ — constant spacing because $g$ is linear.

• Writing a scalar field as though it had components, like $f(x,y,z)=(f_1,f_2,f_3)$ — a scalar field outputs one number, not three; that structure belongs to a vector field.
• Treating the "level surface" and the "graph of $f$" as the same object. The graph of $w=f(x,y,z)$ needs four dimensions; the level surface lives in the same three-dimensional space as the domain, at one fixed output value.
• Forgetting that level surfaces for different constants $c_1\ne c_2$ can never cross — a crossing point would force $f$ to take two different values simultaneously, which contradicts $f$ being a function.
• Assuming every level-surface family looks like nested spheres. A linear field gives parallel planes with constant spacing; only fields like $x^2+y^2+z^2$ give surfaces that bunch up or spread out.

Ask the tutor: Want to see how the level surfaces of $x^2-y^2$ differ from those of $x^2+y^2$, or check whether two given surfaces could belong to the same field?