Why Look Near an Excluded Point?
A field formula may be defined throughout a region except at one point. Although the field has no value at that point, its values nearby can still reveal a recognizable pattern: scalar values may grow, vector arrows may lengthen, or their directions may change as they approach the gap.
These nearby values matter when sketching the field. They help describe what the field does around the missing point, but they do not supply a value at it. In particular, an undefined value is not the same as a zero value.
For a point \((x,y)\) near an excluded point \((a,b)\), its distance from that point is \(r=\sqrt{(x-a)^2+(y-b)^2}\). To investigate the nearby behavior, evaluate the field at several domain points with small \(r\). For a scalar field, compare the resulting values. For a vector field, compare both the arrow lengths, given by vector magnitudes, and the arrow directions.
A Practical Method for Sketching
Choose sample points at different distances and, when useful, in different directions. Exact values are especially helpful: they make it easier to compare how a field changes as the distance shrinks. A sample table can organize the work, but any sketch should reflect the field’s actual domain: do not draw a field value at a point where the formula is undefined.
Identify where the formula fails and confirm that your sample points lie in the domain.
Use points at different distances from the excluded point; sample more than one direction if direction may matter.
For vector fields, calculate arrow lengths as well as components. For scalar fields, track whether values rise, fall, or change with direction.
Show nearby behavior, and leave the excluded point without a field value.
Sampling is a guide to a sketch, not a proof about every nearby point. A few calculations can reveal a likely pattern, but a general formula or argument is needed to establish that the pattern holds throughout a neighborhood.
Worked Example: A Scalar Field That Grows Near the Origin
Worked Example: A Scalar Field That Grows Near the Origin
Consider \(f(x,y)=\frac{1}{x^2+y^2}\), whose domain excludes \((0,0)\). Evaluate it at several points near the origin and describe the values.
At \((1/2,0)\), the denominator is \((1/2)^2+0^2=1/4\), so \(f(1/2,0)=1/(1/4)=4\). The distance to the origin is \(1/2\).
At \((1,0)\), the denominator is \(1^2+0^2=1\), so \(f(1,0)=1\). The distance is \(1\).
At \((0,2)\), the denominator is \(0^2+2^2=4\), so \(f(0,2)=1/4\). The distance is \(2\).
The formula depends on distance squared: at any point a distance \(r>0\) from the origin, \(x^2+y^2=r^2\), so the value is \(1/r^2\). The samples agree: smaller distances give larger values. At the origin the denominator is zero, so the formula is undefined there; it does not take the value \(+\infty\). A sketch should show increasing scalar values toward the gap and no value at the gap.
Check: Substitution gives \(f(1/2,0)=4\), \(f(1,0)=1\), and \(f(0,2)=1/4\), matching the displayed evaluations.
Vector Fields: Length and Direction Both Matter
A vector arrow has both a direction and a magnitude. Near an excluded point, these features may behave differently: the arrow length may grow while the direction turns around the point. To sketch the field accurately, calculate both rather than inferring arrow length from a single component.
Worked Example: Arrows Circling an Excluded Point
Worked Example: Arrows Circling an Excluded Point
Consider \(\mathbf{F}(x,y)=\left\langle\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right\rangle\), defined everywhere except at the origin. Evaluate it at points on the coordinate axes.
At \((1,0)\), the denominator is \(1\), and the numerator components are \(0\) and \(1\). Thus \(\mathbf{F}(1,0)=\langle0,1\rangle\), an upward arrow of magnitude \(\sqrt{0^2+1^2}=1\).
At \((0,1)\), the denominator is \(1\), and the numerator components are \(-1\) and \(0\). Thus \(\mathbf{F}(0,1)=\langle-1,0\rangle\), a leftward arrow of magnitude \(1\).
At \((-1,0)\), the denominator is \(1\), giving \(\mathbf{F}(-1,0)=\langle0,-1\rangle\), a downward arrow of magnitude \(1\). At \((0,-1)\), it gives \(\mathbf{F}(0,-1)=\langle1,0\rangle\), a rightward arrow of magnitude \(1\). These samples suggest a circulation around the origin.
Now compare distances along the positive \(x\)-axis. At \((1/2,0)\), the denominator is \(1/4\) and the numerator is \(\langle0,1/2\rangle\), so \(\mathbf{F}(1/2,0)=\langle0,2\rangle\), with magnitude \(2\). At \((2,0)\), the denominator is \(4\) and the numerator is \(\langle0,2\rangle\), so \(\mathbf{F}(2,0)=\langle0,1/2\rangle\), with magnitude \(1/2\).
For any \((x,y)\ne(0,0)\), the magnitude satisfies \(\|\mathbf{F}(x,y)\|=\frac{\sqrt{(-y)^2+x^2}}{x^2+y^2}=\frac{\sqrt{x^2+y^2}}{x^2+y^2}=\frac{1}{\sqrt{x^2+y^2}}\). Thus the magnitude is \(1/r\), where \(r\) is the distance to the origin. This proves that the arrows grow longer as the distance decreases, while the axis samples show that their directions circulate. At the origin the formula is undefined, so no arrow belongs there.
Check: The computed axis values have magnitudes \(1,1,1,1\) at distance \(1\), magnitude \(2\) at distance \(1/2\), and magnitude \(1/2\) at distance \(2\), consistent with \(1/r\).
Undefined Does Not Mean Zero
A zero vector is a particular value, \(\langle0,0\rangle\). To say a field has that value at a point, the point must be in its domain and the field must be defined there. If a formula has a zero denominator, it gives no value at that point; replacing the missing value by zero changes the field’s definition.
Worked Example: A Missing Value Versus a Defined Zero
Worked Example: A Missing Value Versus a Defined Zero
Compare \(\mathbf{U}(x,y)=\left\langle\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right\rangle\), defined away from the origin, with \(\mathbf{V}(x,y)=\langle x,y\rangle\), defined on the whole plane.
For \(\mathbf{U}\), at \((1,0)\) the denominator is \(1\), so \(\mathbf{U}(1,0)=\langle1,0\rangle\). At \((0,1)\) the denominator is also \(1\), so \(\mathbf{U}(0,1)=\langle0,1\rangle\). Both vectors have magnitude \(1\), since \(\sqrt{1^2+0^2}=1\) and \(\sqrt{0^2+1^2}=1\). But at \((0,0)\) the denominator is zero, so \(\mathbf{U}(0,0)\) is undefined. The different sample directions also show that the nearby vectors do not all point in one direction.
For \(\mathbf{V}\), direct substitution gives \(\mathbf{V}(0,0)=\langle0,0\rangle\), a defined zero vector. At \((1/10,0)\), \(\mathbf{V}(1/10,0)=\langle1/10,0\rangle\), whose magnitude is \(1/10\). In general, \(\|\mathbf{V}(x,y)\|=\sqrt{x^2+y^2}\), the distance from the origin, so these vectors get shorter as their points get closer to the origin.
Check: Substituting the origin into \(\mathbf{U}\) gives a zero denominator, while substituting it into \(\mathbf{V}\) gives \(\langle0,0\rangle\). The first field is undefined there; the second is defined and zero.
A Common Mistake: Assigning a Value to a Gap
For \(\mathbf{U}\) above, a tempting but incorrect calculation is “at the origin, both numerators are zero, so \(\mathbf{U}(0,0)=\langle0,0\rangle\).” That calculation ignores the denominators: each component at the origin has the form \(0/0\), which is undefined. The correction is to state that the formula does not define \(\mathbf{U}\) at the origin. A sketch may show what the arrows do nearby, but it must not place a zero arrow at the excluded point.
A second caution is that a few samples do not establish a complete description of a field. Axis samples can help reveal a pattern, but other directions may behave differently. Use the formula to check what the samples suggest, and distinguish a proven general pattern from a visual guess.
How to Read the Resulting Sketch
Near an excluded point, keep three questions separate:
- Is the field defined at the point? Check the domain, not just the nearby arrows or values.
- What happens to the magnitude or scalar value? Compare exact evaluations or derive a formula in terms of distance.
- What happens to direction? Sample more than one direction when working with vectors.
These observations give a disciplined way to draw the neighborhood of a gap. A scalar field may rise without bound, a vector field may grow while turning, or a defined field may have a zero value at the point. The correct description depends on the formula and its domain.
Check Your Understanding
Use the definitions and examples above to answer the following questions.
- What makes an excluded point isolated?
- For \(f(x,y)=1/(x^2+y^2)\), what are the values at \((1,0)\) and \((0,2)\), and what do they suggest about distance from the origin?
- Why does a zero denominator indicate an undefined value rather than a value of zero?
- For the field \(\mathbf{F}\) in the circulation example, what is its magnitude at a point a distance \(r>0\) from the origin?
- Why can samples along the coordinate axes suggest a vector-field pattern but not, by themselves, prove its behavior in every direction?