Putting It Together: Vector Calculus

Drawing a Rotational Vector Field

A photograph of a hurricane, showing the rotation around its eye.

Figure 1.

Sketch the vector field ${\bf{F}}(x,y)=\langle y,-x\rangle$

Solution

Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors.

$(x,y)$	${\bf{F}}(x,y)$	$(x,y)$	${\bf{F}}(x,y)$	$(x,y)$	${\bf{F}}(x,y)$
$(1,0)$	$\langle 0,-1\rangle$	$(2,0)$	$\langle 0,-2 \rangle$	$(1,1)$	$\langle 1,-1\rangle$
$(0,1)$	$\langle 1,0\rangle$	$(0,2)$	$\langle 2,0\rangle$	$(-1,1)$	$\langle 1,1\rangle$
$(-1,0)$	$\langle 0,1\rangle$	$(-2,0)$	$\langle 0,2\rangle$	$(-1,-1)$	$\langle -1,1\rangle$
$(0,-1)$	$\langle -1,0\rangle$	$(0,-2)$	$\langle -2,0\rangle$	$(1,-1)$	$\langle -1,-1\rangle$

A visual representation of the given vector field in a coordinate plane with two additional diagrams with notation. The first representation shows the vector field. The arrows are circling the origin in a clockwise motion. The second representation shows concentric circles, highlighting the radial pattern. The The third representation shows the concentric circles. It also shows arrows for the radial vector <a,b> for all points (a,b). Each is perpendicular to the arrows in the given vector field.

Figure 2. (a) A visual representation of vector field ${\bf{F}}(x,y)=\langle y,-x\rangle$ . (b) Vector field ${\bf{F}}(x,y)=\langle y,-x\rangle$ with circles centered at the origin. (c) Vector ${\bf{F}}(a,b)$ is perpendicular to radial vector $\langle a,b\rangle$ at point $(a,b)$ .

Analysis

Note that vector ${\bf{F}}(a,b)=\langle b,-a\rangle$ points clockwise and is perpendicular to radial vector $\langle a,b\rangle$ . (We can verify this assertion by computing the dot product of the two vectors: $\langle a,b\rangle\cdot\langle -b,a\rangle= -ab+ab=0$ .) Furthermore, vector $\langle b,-a\rangle$ has length $r=\sqrt{a^{2}+b^{2}}$ . Thus, we have a complete description of this rotational vector field: the vector associated with point $(a,b)$ is the vector with length $r$ tangent to the circle with radius $r$ , and it points in the clockwise direction.

Sketches such as that in Figure 6 under Example “Sketching a Vector Field” are often used to analyze major storm systems, including hurricanes and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an effect caused by Earth’s rotation about its axis and is called the Coriolis Effect.)

Module 6: Vector Calculus