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    13.1 Exercises

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    nope
    This document contains problems involving vector calculus concepts like sketching and analyzing vector fields, finding gradient vector fields, plotting flow lines, and solving differential equations describing particle motion in vector fields. The problems cover topics like matching vector fields to plots, finding points where vector fields are zero, relating gradient vector fields to contour maps, and deriving and solving differential equations for flow lines and particle paths in various vector fields.

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    13.1 Exercises

    Uploaded by

    nope
    34 views2 pages
    This document contains problems involving vector calculus concepts like sketching and analyzing vector fields, finding gradient vector fields, plotting flow lines, and solving differential equations describing particle motion in vector fields. The problems cover topics like matching vector fields to plots, finding points where vector fields are zero, relating gradient vector fields to contour maps, and deriving and solving differential equations for flow lines and particle paths in various vector fields.

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    (13)13.1 Exercises

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    This document contains problems involving vector calculus concepts like sketching and analyzing vector fields, finding gradient vector fields, plotting flow lines, and solving differential equations describing particle motion in vector fields. The problems cover topics like matching vector fields to plots, finding points where vector fields are zero, relating gradient vector fields to contour maps, and deriving and solving differential equations for flow lines and particle paths in various vector fields.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    34 views2 pages

    13.1 Exercises

    Uploaded by

    nope
    This document contains problems involving vector calculus concepts like sketching and analyzing vector fields, finding gradient vector fields, plotting flow lines, and solving differential equations describing particle motion in vector fields. The problems cover topics like matching vector fields to plots, finding points where vector fields are zero, relating gradient vector fields to contour maps, and deriving and solving differential equations for flow lines and particle paths in various vector fields.

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    © All Rights Reserved
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    Download as docx, pdf, or txt
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    #pg766#pgVector Calculus#pg761#pgT1#pg

    #@#1-10 Sketch the vector field \(\mathbf{F}\) by drawing a diagram like Figure 4 or Figure 8 .
    1. \(\mathbf{F}(x, y)=0.3 \mathbf{i}-0.4 \mathbf{j}\)
    2. \(\mathbf{F}(x, y)=\frac{1}{2} x \mathbf{i}+y \mathbf{j}\)
    3. \(\mathbf{F}(x, y)=-\frac{1}{2} \mathbf{i}+(y-x) \mathbf{j}\)
    4. \(\mathbf{F}(x, y)=y \mathbf{i}+(x+y) \mathbf{j}\)
    5. \(\mathbf{F}(x, y)=\frac{y \mathbf{i}+x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}\)
    6. \(\mathbf{F}(x, y)=\frac{y \mathbf{i}-x \mathbf{j}}{\sqrt{x^{2}+y^{2}}}\)
    7. \(\mathbf{F}(x, y, z)=\mathbf{k}\)
    8. \(\mathbf{F}(x, y, z)=-y \mathbf{k}\)
    9. \(\mathbf{F}(x, y, z)=x \mathbf{k}\)
    10. \(\mathbf{F}(x, y, z)=\mathbf{j}-\mathbf{i}\)
    #@#11-14 Match the vector fields \(\mathbf{F}\) with the plots labeled I-IV. Give reasons for your choices.
    11. \(\mathbf{F}(x, y)=\langle x,-y\rangle\)
    12. \(\mathbf{F}(x, y)=\langle y, x-y\rangle\)
    13. \(\mathbf{F}(x, y)=\langle y, y+2\rangle\)
    14. \(\mathbf{F}(x, y)=\langle\cos (x+y), x\rangle\)
    #@#15-18 Match the vector fields \(\mathbf{F}\) on \(\mathbb{R}^{3}\) with the plots labeled I-IV. Give reasons for your choices.
    15. \(\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\)
    16. \(\mathbf{F}(x, y, z)=\mathbf{i}+2 \mathbf{j}+z \mathbf{k}\)
    17. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+3 \mathbf{k}\)
    18. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\)
    19. If you have a CAS that plots vector fields (the command is fieldplot in Maple and PlotvectorField or VectorPlot in Mathematica),
    use it to plot
    \[\mathbf{F}(x, y)=\left(y^{2}-2 x y\right) \mathbf{i}+\left(3 x y-6 x^{2}\right) \mathbf{j}\]
    Explain the appearance by finding the set of points \((x, y)\) such that \(\mathbf{F}(x, y)=\mathbf{0}\).
    20. Let \(\mathbf{F}(\mathbf{x})=\left(r^{2}-2 r\right) \mathbf{x}\), where \(\mathbf{x}=\langle x, y\rangle\) and \(r=|\mathbf{x}|\).
    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 \(\mathbf{F}(\mathbf{x})=\mathbf{0}\)
    #@#21-24 Find the gradient vector field of \(f\).
    21. \(f(x, y)=x e^{x y}\)
    22. \(f(x, y)=\tan (3 x-4 y)\)
    23. \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\)
    24. \(f(x, y, z)=x \ln (y-2 z)\)
    #pg767#pg
    #@#25-26 Find the gradient vector field \(\nabla f\) of \(f\) and sketch it.
    25. \(f(x, y)=x^{2}-y\)
    26. \(f(x, y)=\sqrt{x^{2}+y^{2}}\)
    27-28 " Plot the gradient vector field of \(f\) together with a contour map of \(f\). Explain how they are related to each other.
    27. \(f(x, y)=\ln \left(1+x^{2}+2 y^{2}\right)\)
    28. \(f(x, y)=\cos x-2 \sin y\)
    29. A particle moves in a velocity field \(\mathbf{V}(x, y)=\left\langle x^{2}, x+y^{2}\right\rangle .\) If it is at position \((2,1)\) at
    time \(t=3\), estimate its location at time \(t=3.01\)
    30. At time \(t=1\), a particle is located at position \((1,3) .\) If it moves in a velocity field
    \[\mathbf{F}(x, y)=\left\langle x y-2, y^{2}-10\right\rangle\]
    find its approximate location at time \(t=1.05\).
    31. The flow lines (or 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 \(\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}\) 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 \(d x / d
    t=x\) and \(d y / d t=-y .\) Then solve the differential equations to find an equation of the flow line that passes through the point \
    ((1,1)\).
    32.
    a. Sketch the vector field \(\mathbf{F}(x, y)=\mathbf{i}+x \mathbf{j}\) 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 \(d y / d x=x\)
    c. If a particle starts at the origin in the velocity field given by \(\mathbf{F}\), find an equation of the path it follows.

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