From a Scalar Field to a Surface
A scalar field of two variables assigns a real number to each point in its domain. When we write \(z=f(x,y)\), the inputs are the two coordinates \(x\) and \(y\), and the output \(f(x,y)\) is represented by the vertical coordinate \(z\). The graph collects these inputs and outputs into a surface in three-dimensional space.
This gives a geometric way to interpret a formula: choose a location \((x,y)\) in the input plane, evaluate the field there, and use the result as the height above that location. A positive output places the graph above the \(xy\)-plane, a negative output places it below, and an output of zero places it on that plane.
The graph is not just the collection of input points \((x,y)\), nor is the output a third independent input. The formula determines the third coordinate from the first two. In a sketch, the surface may be pictured as rising or falling above the plane of possible inputs.
Why the Graph Has the Height Given by the Formula
The definition follows directly from what a function does. For every allowed input \((x,y)\), the function assigns the output \(f(x,y)\). Placing that output in the third coordinate gives a point \((x,y,f(x,y))\). Conversely, any point on the graph must have this form, because its third coordinate is required to equal the function value at its first two coordinates.
This also explains a useful check when evaluating a field: once the height has been calculated, the corresponding point on the surface must retain the original \(x\)- and \(y\)-coordinates. Only the third coordinate is replaced by the output.
Worked Example: A Bowl-Shaped Surface
Worked Example: A Bowl-Shaped Surface
Consider \(f(x,y)=x^2+y^2\). Find the surface heights at \((0,0)\), \((1,2)\), and \((-2,1)\), and write the corresponding points on the graph.
At \((0,0)\), substitute \(x=0\) and \(y=0\):
The graph point is \((0,0,0)\). Checking its height gives \(z=0^2+0^2=0\), as required.
At \((1,2)\):
The graph point is \((1,2,5)\). Substitution into the surface equation gives \(z=1^2+2^2=5\), so the point lies on the graph.
At \((-2,1)\):
The graph point is \((-2,1,5)\). Checking gives \(z=(-2)^2+1^2=4+1=5\), confirming the height.
Both of the last two inputs have height \(5\), even though they are different locations in the input plane. The squared terms are nonnegative, so this surface never has a height below zero. As either coordinate moves farther from zero while the other is fixed, its squared contribution increases. The resulting surface has a low point at the origin and rises away from it, giving the bowl-like shape.
Worked Example: A Saddle-Shaped Surface
Worked Example: A Saddle-Shaped Surface
Let \(g(x,y)=x^2-y^2\). Find the heights at \((0,0)\), \((2,1)\), and \((1,2)\), and interpret their signs.
At the origin:
The corresponding graph point is \((0,0,0)\), and \(0=0^2-0^2\), so it satisfies the equation.
At \((2,1)\):
The graph point is \((2,1,3)\), and checking gives \(3=2^2-1^2=4-1\).
At \((1,2)\):
The graph point is \((1,2,-3)\), and checking gives \(-3=1^2-2^2\).
The signs show that the surface rises above the \(xy\)-plane at \((2,1)\) and lies below it at \((1,2)\). In the direction where \(y=0\), the formula becomes \(g(x,0)=x^2\), which is nonnegative and increases as \(|x|\) grows. In the direction where \(x=0\), it becomes \(g(0,y)=-y^2\), which is nonpositive and decreases as \(|y|\) grows. The surface therefore rises in one direction and falls in another. This opposing behavior produces a saddle shape rather than a bowl.
Reading the Signs and Terms in a Formula
For simple formulas, the terms often suggest how the surface behaves in different directions. In \(x^2+y^2\), both squared terms contribute positively to the height. In \(x^2-y^2\), one squared term contributes positively and the other negatively. Looking at the formula is not a substitute for evaluating particular points, but it can guide a sketch by indicating which directions tend to rise and which tend to fall.
The coefficients also matter. A larger positive coefficient makes the corresponding squared contribution grow more quickly as that coordinate changes. For example, for \(h(x,y)=4x^2+y^2\), moving from \((0,0)\) to \((1,0)\) gives height \(4\), while moving to \((0,1)\) gives height \(1\). These values show that equal changes in the two coordinate directions need not produce equal changes in height.
Worked Example: A Tilted Plane
Worked Example: A Tilted Plane
Consider \(p(x,y)=2x-y+1\). Find the graph heights at \((0,0)\), \((1,0)\), \((0,1)\), and \((2,-1)\). Then check each resulting three-dimensional point.
At \((0,0)\):
The graph point is \((0,0,1)\). Substituting its first two coordinates into the formula gives \(z=1\), as in the point.
At \((1,0)\):
The graph point is \((1,0,3)\), and \(2(1)-0+1=3\).
At \((0,1)\):
The graph point is \((0,1,0)\), and \(2(0)-1+1=0\).
At \((2,-1)\):
The graph point is \((2,-1,6)\), and \(2(2)-(-1)+1=6\).
The constant term gives height \(1\) at the origin. Increasing \(x\) by \(1\) while holding \(y=0\) increases the height from \(1\) to \(3\). Increasing \(y\) by \(1\) while holding \(x=0\) decreases it from \(1\) to \(0\). Since the formula is linear in \(x\) and \(y\), its graph is a tilted plane. The coefficient of \(x\) is positive, while the coefficient of \(y\) is negative, so these coordinate directions affect height in opposite ways.
A Reliable Method for Plotting Sample Points
When making a rough sketch from a formula, a small table of input points and calculated heights is often a good starting point. Keep the inputs and outputs organized: each row should show an input pair, its field value, and the corresponding point on the surface.
Select locations \((x,y)\) in the domain of the field.
Substitute both input coordinates carefully to obtain \(f(x,y)\).
Record \((x,y,f(x,y))\), preserving the original inputs as the first two coordinates.
Substitute its first two coordinates into \(z=f(x,y)\) and confirm that the result matches its third coordinate.
A finite selection of points helps reveal the surface’s general form, but it does not specify every point on the graph. A drawing connects the sampled information in a way consistent with the formula. It should not suggest a sharp edge, a missing point, or behavior outside the domain unless the formula and domain support that feature.
A Common Mistake: Losing a Sign
When a coordinate is negative, squaring it does not leave it negative. For the bowl \(f(x,y)=x^2+y^2\), a mistaken calculation might say:
This treats the negative sign as if it remained outside the square. The correct substitution groups the coordinate before squaring:
The correction changes the height from \(-3\) to \(5\), and therefore changes the graph point from an incorrect \((-2,1,-3)\) to the correct \((-2,1,5)\). Checking the point against the original formula catches the mistake: the third coordinate must equal \((-2)^2+1^2=5\).
Why Height Surfaces Matter
Thinking of a scalar field as a height surface gives a direct visual interpretation to its numerical values. The input plane records where the field is evaluated; the vertical coordinate records what value it assigns there. This viewpoint applies across many formulas: a bowl can represent values that grow in all directions from a low point, a saddle can rise in one direction and fall in another, and a plane can change height steadily in different coordinate directions.
The central discipline is simple: evaluate the formula at an input, then use that output as the graph’s height above the same input. Careful substitution and a final check make the connection between algebra and geometry precise.
Check Your Understanding
For each question, keep the input coordinates distinct from the height.
- For \(f(x,y)=x^2+y^2\), what is the graph point above \((2,-1)\)?
- For \(g(x,y)=x^2-y^2\), which of the points \((2,1)\) and \((1,2)\) lies below the \(xy\)-plane on the graph?
- For \(p(x,y)=2x-y+1\), what is the height above \((1,1)\)?
- Explain why \((x,y,f(x,y))\), rather than \((f(x,y),x,y)\), is the graph point under the convention \(z=f(x,y)\).
- What sign error can occur when evaluating a squared coordinate that is negative, and how can you avoid it?