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16.1E: Mazoezi ya Sehemu ya 16.1

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  • Page ID
    178934

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    1. Uwanja wa uwanja wa vector\(\vecs{F}=\vecs{F}(x,y)\) ni seti ya pointi\((x,y)\) katika ndege, na aina mbalimbali\(\vecs F\) ni seti ya nini katika ndege?

    Jibu
    Vectors

    Kwa mazoezi 2 - 4, onyesha kama taarifa hiyo ni ya kweli au ya uongo.

    2. Shamba la Vector\(\vecs{F}=⟨3x^2,1⟩\) ni shamba la gradient kwa wote\(ϕ_1(x,y)=x^3+y\) na\(ϕ_2(x,y)=y+x^3+100.\)

    3. Shamba la Vector\(\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}\) ni mara kwa mara katika mwelekeo na ukubwa kwenye mduara wa kitengo.

    Jibu
    Uongo

    4. Shamba la Vector\(\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}\) sio shamba la radial wala shamba la mzunguko.

    Kwa mazoezi ya 5 - 13, kuelezea kila shamba la vector kwa kuchora baadhi ya vectors zake.

    5. [T]\(\vecs{F}(x,y)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)

    Jibu

    clipboard_ed15464787a3931076aab028ddc5b12d7.png

    6. [T]\(\vecs{F}(x,y)=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\)

    7. [T]\(\vecs{F}(x,y)=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}\)

    Jibu

    clipboard_ec0907b086839ad5c262019c65af7add7.png

    8. [T]\(\vecs{F}(x,y)=\,\hat{\mathbf i}+\,\hat{\mathbf j}\)

    9. [T]\(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}\)

    Jibu

    clipboard_ef331d6cd76b60efd3886d0df3b197637.png

    10. [T] \(\vecs{F}(x,y)=3\,\hat{\mathbf i}+x\,\hat{\mathbf j}\)

    11. [T]\(\vecs{F}(x,y)=y\,\hat{\mathbf i}+\sin x\,\hat{\mathbf j}\)

    Jibu

    clipboard_e90f52faea8df656c5e8538475924d4a3.png

    12. [T]\(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}+z\,\hat{\mathbf k}\)

    13. [T]\(\vecs F(x,y,z)=2x\,\hat{\mathbf i}−2y\,\hat{\mathbf j}−2z\,\hat{\mathbf k}\)

    Jibu

    clipboard_e1eb143e779ce49ea2a01af87d7a3548d.png

    14. [T]\(\vecs F(x,y,z)=yz\,\hat{\mathbf i}−xz\,\hat{\mathbf j}\)

    Kwa mazoezi 15 - 20, pata shamba la vector gradient la kila kazi\(f\).

    15. \(f(x,y)=x\sin y+\cos y\)

    Jibu
    \(\vecs{F}(x,y)=\sin(y)\,\hat{\mathbf i}+(x\cos y−\sin y)\,\hat{\mathbf j}\)

    16. \(f(x,y,z)=ze^{−xy}\)

    17. \(f(x,y,z)=x^2y+xy+y^2z\)

    Jibu
    \(\vecs F(x,y,z)=(2xy+y)\,\hat{\mathbf i}+(x^2+x+2yz)\,\hat{\mathbf j}+y^2\,\hat{\mathbf k}\)

    18. \(f(x,y)=x^2\sin(5y)\)

    19. \(f(x,y)=\ln(1+x^2+2y^2)\)

    Jibu
    \(\vecs{F}(x,y)=\dfrac{2x}{1+x^2+2y^2}\,\hat{\mathbf i}+\dfrac{4y}{1+x^2+2y^2}\,\hat{\mathbf j}\)

    20. \(f(x,y,z)=x\cos\left(\frac{y}{z}\right)\)

    21. ni vector shamba\(\vecs{F}(x,y)\) na thamani katika\((x,y)\) kwamba ni ya kitengo urefu na pointi kuelekea\((1,0)\)?

    Jibu
    \(\vecs{F}(x,y)=\dfrac{(1−x)\,\hat{\mathbf i}−y\,\hat{\mathbf j}}{\sqrt{(1−x)^2+y^2}}\)

    Kwa mazoezi 22 - 24, andika formula kwa mashamba ya vector na mali zilizopewa.

    22. Vectors wote ni sawa na\(x\) -axis na wadudu wote kwenye mstari wa wima wana ukubwa sawa.

    23. Vectors wote wanasema kuelekea asili na kuwa na urefu wa mara kwa mara.

    Jibu
    \(\vecs{F}(x,y)=\dfrac{(y\,\hat{\mathbf i}−x\,\hat{\mathbf j})}{\sqrt{x^2+y^2}}\)

    24. Vectors wote ni wa kitengo urefu na ni perpendicular kwa vector nafasi katika hatua hiyo.

    25. Kutoa formula\(\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}\) kwa uwanja wa vector katika ndege ambayo ina mali ambayo\(\vecs{F}=\vecs 0\) katika\((0,0)\) na kwamba katika hatua nyingine yoyote\((a,b), \vecs F\) ni tangent kwa mzunguko\(x^2+y^2=a^2+b^2\) na pointi katika mwelekeo clockwise na ukubwa\(\|\vecs F\|=\sqrt{a^2+b^2}\).

    Jibu
    \(\vecs{F}(x,y)=y\,\hat{\mathbf i}−x\,\hat{\mathbf j}\)

    26. Ni uwanja wa vector shamba\(\vecs{F}(x,y)=(P(x,y),Q(x,y))=(\sin x+y)\,\hat{\mathbf i}+(\cos y+x)\,\hat{\mathbf j}\) la gradient?

    27. Kupata formula kwa ajili ya uwanja vector\(\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}\) kutokana na ukweli kwamba kwa pointi zote\((x,y)\),\(\vecs F\) pointi kuelekea asili na\(\|\vecs F\|=\dfrac{10}{x^2+y^2}\).

    Jibu
    \(\vecs{F}(x,y)=\dfrac{−10}{(x^2+y^2)^{3/2}}(x\,\hat{\mathbf i}+y\,\hat{\mathbf j})\)

    Kwa mazoezi 28 - 29, kudhani kwamba uwanja wa umeme katika\(xy\) -ndege unasababishwa na mstari usio wa malipo kando ya\(x\) mhimili ni shamba la gradient na kazi ya uwezo\(V(x,y)=c\ln\left(\frac{r_0}{\sqrt{x^2+y^2}}\right)\), ambapo\(c>0\)\(r_0\) ni mara kwa mara na ni umbali wa kumbukumbu ambayo uwezo ni kudhani kuwa sifuri.

    28. Find sehemu ya uwanja wa umeme katika\(x\) - na\(y\) -maelekezo, ambapo\(\vecs E(x,y)=−\vecs ∇V(x,y).\)

    29. Onyesha kwamba shamba la umeme kwa hatua katika\(xy\) -ndege linaelekezwa nje kutoka kwa asili na ina ukubwa\(\|\vecs E\|=\dfrac{c}{r}\), wapi\(r=\sqrt{x^2+y^2}\).

    Jibu
    \(\|\vecs E\|=\dfrac{c}{|r|^2}r=\dfrac{c}{|r|}\dfrac{r}{|r|}\)

    mstari kati yake (au kuboresha) ya uwanja vector\(\vecs F\) ni Curve\(\vecs r(t)\) vile kwamba\(d\vecs{r}/dt=\vecs F(\vecs r(t))\). Ikiwa\(\vecs F\) inawakilisha uwanja wa kasi wa chembe inayohamia, basi mistari ya mtiririko ni njia zilizochukuliwa na chembe. Kwa hiyo, mistari ya mtiririko ni tangent kwenye uwanja wa vector.

    Kwa mazoezi 30 na 31, onyesha kuwa Curve iliyotolewa\(\vecs c(t)\) ni mstari wa mtiririko wa shamba la vector la kasi lililopewa\(\vecs F(x,y,z)\).

    30. \(\vecs c(t)=⟨ e^{2t},\ln|t|,\frac{1}{t} ⟩,\,t≠0;\quad \vecs F(x,y,z)=⟨2x,z,−z^2⟩\)

    31. \(\vecs c(t)=⟨ \sin t,\cos t,e^t⟩;\quad \vecs F(x,y,z) =〈y,−x,z〉\)

    Jibu
    \(\vecs c′(t)=⟨ \cos t,−\sin t,e^{−t}⟩=\vecs F(\vecs c(t))\)

    Kwa mazoezi 32 - 34, basi\(\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)\(\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\), na\(\vecs H=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}\). mechi\(\vecs F\),\(\vecs G\),\(\vecs H\) na kwa grafu zao.

    32.

    clipboard_ed92420bb9a2a79bf7e0737bdf653eee9.png

    33.

    clipboard_eb5a596effc35031ee81697fe6f857207.png

    Jibu
    \(\vecs H\)

    34.

    clipboard_ec5488ed5d7cc242a516fe39456391eaa.png

    Kwa mazoezi 35 - 38, basi\(\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\),\(\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\), na\(\vecs H=−x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\). Mechi ya mashamba ya vector na grafu zao katika (I) - (IV).

    1. \(\vecs F+\vecs G\)
    2. \(\vecs F+\vecs H\)
    3. \(\vecs G+\vecs H\)
    4. \(−\vecs F+\vecs G\)

    35.

    clipboard_eddbf3f9c662003d8e6e632e520b53134.png

    Jibu
    d.\(−\vecs F+\vecs G\)

    36.

    clipboard_e75a38125572207465d05c29854873433.png

    37.

    clipboard_e0404b6b216ac76764fdd9b22d61f3e46.png

    Jibu
    a.\(\vecs F+\vecs G\)

    38.

    clipboard_e3db08e2f2eec3aeb28cd303fd451bb14.png