Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 16 ## Vector Calculus ## Educators FL ### Problem 1 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = 0.3 \textbf{i} - 0.4 \textbf{j} $FL Frank L. Numerade Educator ### Problem 2 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = \frac{1}{2}x \textbf{i} + y \textbf{j} $FL Frank L. Numerade Educator ### Problem 3 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = -\frac{1}{2} \textbf{i} + (y - x) \textbf{j} $FL Frank L. Numerade Educator ### Problem 4 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = y \textbf{i} + (x + y) \textbf{j} $FL Frank L. Numerade Educator ### Problem 5 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = \dfrac{y \textbf{i} + x \textbf{j}}{\sqrt{x^2 + y^2}} $FL Frank L. Numerade Educator ### Problem 6 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y) = \dfrac{y \textbf{i} - x \textbf{j}}{\sqrt{x^2 + y^2}} $FL Frank L. Numerade Educator ### Problem 7 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y, z) = \textbf{i} $FL Frank L. Numerade Educator ### Problem 8 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y, z) = z \textbf{i} $FL Frank L. Numerade Educator ### Problem 9 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y, z) = -y \textbf{i} $FL Frank L. Numerade Educator ### Problem 10 Sketch the vector field$ \textbf{F} $by drawing a diagram like Figure 5 or Figure 9.$ \textbf{F} (x, y, z) = \textbf{i} + \textbf{k} $FL Frank L. Numerade Educator ### Problem 11 Match the vector fields$ \textbf{F} $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y) = \langle x, -y \rangle $FL Frank L. Numerade Educator ### Problem 12 Match the vector fields$ \textbf{F} $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y) = \langle y, x - y \rangle $FL Frank L. Numerade Educator ### Problem 13 Match the vector fields$ \textbf{F} $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y) = \langle y, y + 2 \rangle $FL Frank L. Numerade Educator ### Problem 14 Match the vector fields$ \textbf{F} $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y) = \langle \cos (x + y), x \rangle $FL Frank L. Numerade Educator ### Problem 15 Match the vector fields$ \textbf{F} $on$ \mathbb{R}^3 $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y, z) = \textbf{i} + 2\textbf{j} + 3\textbf{k} $FL Frank L. Numerade Educator ### Problem 16 Match the vector fields$ \textbf{F} $on$ \mathbb{R}^3 $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y, z) = \textbf{i} + 2 \textbf{j} + z \textbf{k} $FL Frank L. Numerade Educator ### Problem 17 Match the vector fields$ \textbf{F} $on$ \mathbb{R}^3 $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y, z) = x \textbf{i} + y \textbf{j} + 3 \textbf{k} $FL Frank L. Numerade Educator ### Problem 18 Match the vector fields$ \textbf{F} $on$ \mathbb{R}^3 $with the plots labeled I - IV. Give reasons for your choices.$ \textbf{F} (x, y, z) = x \textbf{i} + y \textbf{j} + z \textbf{k} $FL Frank L. Numerade Educator ### Problem 19 If you have a CAS that plots vector fields (the command is$ \large \text{fieldplot} $in Maple and$ \large \text{PlotVectorField} $or$ \large \text{VectorPlot} $in Mathematica), use it to plot $$\textbf{F} (x, y) = (y^2 - 2xy) \textbf{i} + (3xy - 6x^2) \textbf{j}$$ Explain the appearance by finding the set of points$ (x, y) $such that$ \textbf{F} (x, y) = 0 $. FL Frank L. Numerade Educator ### Problem 20 Let$ \textbf{F(x)} = (r^2 - 2r)\textbf{x} $, where$ \textbf{x} = \langle x, y \rangle $and$ r = | \textbf{x} | $. Use a CAS to plot this vector field in various domains until you can see what is happening. Describe the appearance of the plot and explain it by finding the points where$ \textbf{F(x) = 0} $. FL Frank L. Numerade Educator ### Problem 21 Find the gradient vector field of$ f $.$ f(x, y) = y \sin (xy) $Bobby B. University of North Texas ### Problem 22 Find the gradient vector field of$ f $.$ f(s, t) = \sqrt{2s + 3t} $FL Frank L. Numerade Educator ### Problem 23 Find the gradient vector field of$ f $.$ f(x, y, z) = \sqrt{x^2 + y^2 + z^2} $FL Frank L. Numerade Educator ### Problem 24 Find the gradient vector field of$ f $.$ f(x, y, z) = x^2 y e^{y/z} $Bobby B. University of North Texas ### Problem 25 Find the gradient vector field$ \nabla f $of$ f $and sketch it.$ f(x, y) = \frac{1}{2}(x - y)^2 $FL Frank L. Numerade Educator ### Problem 26 Find the gradient vector field$ \nabla f $of$ f $and sketch it.$ f(x, y) = \frac{1}{2}(x^2 - y^2) $FL Frank L. Numerade Educator ### Problem 27 Plot the gradient vector field of$ f $together with a contour map of$ f $. Explain how they are related to each other.$ f(x, y) = \ln (1 + x^2 + 2y^2) $FL Frank L. Numerade Educator ### Problem 28 Plot the gradient vector field of$ f $together with a contour map of$ f $. Explain how they are related to each other.$ f(x, y) = \cos x - 2 \sin y $Carson M. Numerade Educator ### Problem 29 Match the functions$ f $with the plots of their gradient vector fields labeled I - IV. Give reasons for your choices.$ f(x, y) = x^2 + y^2 $FL Frank L. Numerade Educator ### Problem 30 Match the functions$ f $with the plots of their gradient vector fields labeled I - IV. Give reasons for your choices.$ f(x, y) = x(x + y) $FL Frank L. Numerade Educator ### Problem 31 Match the functions$ f $with the plots of their gradient vector fields labeled I - IV. Give reasons for your choices.$ f(x, y) = (x + y)^2 $FL Frank L. Numerade Educator ### Problem 32 Match the functions$ f $with the plots of their gradient vector fields labeled I - IV. Give reasons for your choices.$ f(x, y) = \sin \sqrt{x^2 + y^2} $FL Frank L. Numerade Educator ### Problem 33 A particle moves in a velocity field$ \textbf{V}(x, y) = \langle x^2, x + y^2 \rangle $. If it is at position$ (2, 1) $at time$ t = 3 $, estimate its location at time$ t = 3.01 $. FL Frank L. Numerade Educator ### Problem 34 At time$ t = 1 $, a particle is located at position$ (1, 3) $. If it moves in a velocity field $$\textbf{F}(x, y) = \langle xy - 2, y^2 - 10 \rangle$$ find its approximate location at time$ t = 1.05 $. FL Frank L. Numerade Educator ### Problem 35 The$ \textbf{flow lines} $(or$ \textbf{streamlines} $) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field$ \textbf{F}(x, y) = x \textbf{i} - y \textbf{j} $to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are$ x = x(t) $,$ y = y(t) $, explain why these functions satisfy the differential equations$ dx/dt = x $and$ dy/dt = -y $. Then solve the differential equations to find an equation of the flow line that passes through the point$ (1, 1) $. FL Frank L. Numerade Educator ### Problem 36 (a) Sketch the vector field$ \textbf{F}(x, y) = \textbf{i} + x \textbf{j} $and then sketch some flow lines. What shape do these flow lines appear to have? (b) If parametric equations of the flow lines are$ x = x(t) $,$ y = y(t) $, what differential equations do these functions satisfy? Deduce that$ dy/dx = x $. (c) If a particle starts at the origin in the velocity field given by$ \textbf{F} \$, find an equation of the path it follows.

FL
Frank L.
Numerade Educator