6.1 Vector fields (Page 3/15)

Calculus volume 3 Page 3 / 15

Chapter opener: drawing a rotational vector field

A photgraph of a hurricane, showing the rotation around its eye. — (credit: modification of work by NASA)

Sketch the vector field $F (x, y) = ⟨ y, − x ⟩ .$

Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors. [link] shows the resulting vector field.

$(x, y)$	$F (x, y)$	$(x, y)$	$F (x, y)$	$(x, y)$	$F (x, y)$
$(1, 0)$	$⟨ 0, −1 ⟩$	$(2, 0)$	$⟨ 0, −2 ⟩$	$(1, 1)$	$⟨ 1, −1 ⟩$
$(0, 1)$	$⟨ 1, 0 ⟩$	$(0, 2)$	$⟨ 2, 0 ⟩$	$(−1, 1)$	$⟨ 1, 1 ⟩$
$(−1, 0)$	$⟨ 0, 1 ⟩$	$(−2, 0)$	$⟨ 0, 2 ⟩$	$(−1, −1)$	$⟨ −1, 1 ⟩$
$(0, −1)$	$⟨ −1, 0 ⟩$	$(0, −2)$	$⟨ −2, 0 ⟩$	$(1, −1)$	$⟨ −1, −1 ⟩$

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. — (a) A visual representation of vector field $F (x, y) = ⟨ y, − x ⟩ .$ (b) Vector field $F (x, y) = ⟨ y, − x ⟩$ with circles centered at the origin. (c) Vector $F (a, b)$ is perpendicular to radial vector $⟨ a, b ⟩$ at point $(a, b) .$

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Sketching a vector field

Sketch vector field $F (x, y) = \frac{y}{x^{2} + y^{2}} i - \frac{x}{x^{2} + y^{2}} j .$

To visualize this vector field, first note that the dot product $F (a, b) \cdot (a i + b j)$ is zero for any point $(a, b) .$ Therefore, each vector is tangent to the circle on which it is located. Also, as $(a, b) \to (0, 0),$ the magnitude of $F (a, b)$ goes to infinity. To see this, note that

| | F (a, b) | | = \sqrt{\frac{a^{2} + b^{2}}{{(a^{2} + b^{2})}^{2}}} = \sqrt{\frac{1}{a^{2} + b^{2}}} .

Since $\frac{1}{a^{2} + b^{2}} \to \infty$ as $(a, b) \to (0, 0),$ then $| | F (a, b) | | \to \infty$ as $(a, b) \to (0, 0) .$ This vector field looks similar to the vector field in [link] , but in this case the magnitudes of the vectors close to the origin are large. [link] shows a sample of points and the corresponding vectors, and [link] shows the vector field. Note that this vector field models the whirlpool motion of the river in [link] (b). The domain of this vector field is all of $ℝ^{2}$ except for point $(0, 0) .$

$(x, y)$	$F (x, y)$	$(x, y)$	$F (x, y)$	$(x, y)$	$F (x, y)$
$(1, 0)$	$⟨ 0, −1 ⟩$	$(2, 0)$	$⟨ 0, - \frac{1}{2} ⟩$	$(1, 1)$	$⟨ \frac{1}{2}, - \frac{1}{2} ⟩$
$(0, 1)$	$⟨ 1, 0 ⟩$	$(0, 2)$	$⟨ \frac{1}{2}, 0 ⟩$	$(−1, 1)$	$⟨ \frac{1}{2}, \frac{1}{2} ⟩$
$(−1, 0)$	$⟨ 0, 1 ⟩$	$(−2, 0)$	$⟨ 0, \frac{1}{2} ⟩$	$(−1, −1)$	$⟨ - \frac{1}{2}, \frac{1}{2} ⟩$
$(0, −1)$	$⟨ −1, 0 ⟩$	$(0, −2)$	$⟨ - \frac{1}{2}, 0 ⟩$	$(1, −1)$	$⟨ - \frac{1}{2}, - \frac{1}{2} ⟩$

A visual representation of the given vector field in a coordinate plane. The magnitude is larger closer to the origin. The arrows are rotating the origin clockwise. It could be use to model whirlpool motion of a fluid. — A visual representation of vector field $F (x, y) = \frac{y}{x^{2} + y^{2}} i - \frac{x}{x^{2} + y^{2}} j .$ This vector field could be used to model whirlpool motion of a fluid.

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Sketch vector field $F (x, y) = ⟨ −2 y, 2 x ⟩ .$ Is the vector field radial, rotational, or neither?

Rotational
A visual representation of a rotational vector field in a coordinate plane. The arrows circle the origin in a counterclockwise manner.

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Velocity field of a fluid

Suppose that $v (x, y) = - \frac{2 y}{x^{2} + y^{2}} i + \frac{2 x}{x^{2} + y^{2}} j$ is the velocity field of a fluid. How fast is the fluid moving at point $(1, −1) ?$ (Assume the units of speed are meters per second.)

To find the velocity of the fluid at point $(1, −1),$ substitute the point into v :

v (1, −1) = - \frac{2 (−1)}{1 + 1} i + \frac{2 (1)}{1 + 1} j = i + j .

The speed of the fluid at $(1, −1)$ is the magnitude of this vector. Therefore, the speed is $| | i + j | | = \sqrt{2}$ m/sec.

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Vector field $v (x, y) = ⟨ 4 | x |, 1 ⟩$ models the velocity of water on the surface of a river. What is the speed of the water at point $(2, 3) ?$ Use meters per second as the units.

$\sqrt{65}$ m/sec

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We have examined vector fields that contain vectors of various magnitudes, but just as we have unit vectors, we can also have a unit vector field. A vector field F is a unit vector field if the magnitude of each vector in the field is 1. In a unit vector field, the only relevant information is the direction of each vector.

A unit vector field

Show that vector field $F (x, y) = ⟨ \frac{y}{\sqrt{x^{2} + y^{2}}}, - \frac{x}{\sqrt{x^{2} + y^{2}}} ⟩$ is a unit vector field.

To show that F is a unit field, we must show that the magnitude of each vector is 1. Note that

\begin{matrix} \sqrt{{(\frac{y}{\sqrt{x^{2} + y^{2}}})}^{2} + {(- \frac{x}{\sqrt{x^{2} + y^{2}}})}^{2}} & = \sqrt{\frac{y^{2}}{x^{2} + y^{2}} + \frac{x^{2}}{x^{2} + y^{2}}} \\ = \sqrt{\frac{x^{2} + y^{2}}{x^{2} + y^{2}}} \\ = 1. \end{matrix}

Therefore, F is a unit vector field.

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Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

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