Stewart Precalc 6e Section 9.FOM: Focus on Modeling - Vector Fields

Sketch the vector field \(\mathbf{F}(x, y) = \dfrac{1}{2} \mathbf{i} + \dfrac{1}{2} \mathbf{j}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y) = \mathbf{i} + x \mathbf{j}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y) = y \mathbf{i} + \dfrac{1}{2} \mathbf{j}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y) = (x - y) \mathbf{i} + x \mathbf{j}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y) = \dfrac{y \mathbf{i} + x \mathbf{j}}{\sqrt{x^2 + y^2}}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y) = \dfrac{y \mathbf{i} - x \mathbf{j}}{\sqrt{x^2 + y^2}}\) by drawing a diagram as in Figure 3.

Sketch the vector field \(\mathbf{F}(x, y, z) = \mathbf{j}\) by drawing a diagram as in Figure 5.

Sketch the vector field \(\mathbf{F}(x, y, z) = \mathbf{i} - \mathbf{k}\) by drawing a diagram as in Figure 5.

Sketch the vector field \(\mathbf{F}(x, y, z) = z \mathbf{j}\) by drawing a diagram as in Figure 5.

Sketch the vector field \(\mathbf{F}(x, y, z) = y \mathbf{k}\) by drawing a diagram as in Figure 5.

Match the vector field \(\mathbf{F}(x, y) = \langle y, x \rangle\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y) = \langle 1, \sin y \rangle\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y) = \langle x - 2, y + 1 \rangle\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y) = \langle y, \dfrac{1}{x} \rangle\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y, z) = \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y, z) = \mathbf{i} + 2 \mathbf{j} + z \mathbf{k}\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} + 3 \mathbf{k}\) with the graphs labeled I-IV.

Match the vector field \(\mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\) with the graphs labeled I-IV.

The current in a turbulent bay is described by the velocity vector field \(\mathbf{F}(x, y) = (x + y) \mathbf{i} + (x - y) \mathbf{j}\). A small toy boat placed in this bay follows a path called a flow line (or streamline) of the vector field. A streamline starting at \((1, -3)\) is shown in blue in the figure. Sketch streamlines starting at each of the following points: (a) \((1, 4)\), (b) \((-2, 1)\), (c) \((-1, -2)\).

Graph the vector field \(\mathbf{F}(x, y) = x \mathbf{i} + y \mathbf{j}\). What does the graph indicate?

A potter's wheel has a radius of 5 inches. The velocity of each point on the wheel is given by the vector field \(\mathbf{F}(x, y) = -y \mathbf{i} + x \mathbf{j}\). What does the graph indicate?

Graph the vector field \(\mathbf{F}(x, y, z) = z \mathbf{k}\). What does the graph indicate?