速度场：势场可视化

几何画板及绘图内容加载中，请稍等……

2016-11-05 16:29

①本范例演示了速度场中势场的图形化表示，通过动态展示向量场和势函数的关系揭示流体力学或物理场中的核心概念；②涉及的核心数学概念包括向量场、势函数、导数、极限和微分方程；③作图要点运用了Derivative（导数计算）、Sequence（序列生成）、Textfield（文本输入）、Plane（平面定义）等GeoGebra命令和对象；④适合大学物理、工程力学或高等数学课程的教学演示与学习探索。

GeoGebra 命令及参数

Derivative((-Pot(x, y)), x) → F
Derivative((-Pot(x, y)), y) → G
Sequence((x(Element[A, i]), y(Element[A, i]), Pot(x(Element[A, i]), y(Element[A, i]))), i, 1, n) → ZA
Sequence(Sequence[Vector[(j, k), (j + (scale * F(j, k)), k + (scale * G(j, k)))], j, xmin, xmax, xstep], k, ymin, ymax, ystep) → field, {{(0.050768254456645234, 0.04182611872848607), (0.05163855865332412, 0.054698310911191594), (0.04795301573206068, 0.07111214264811228), (0.035967139113809665, 0.08889605405969991), (0.013702414243129124, 0.10160028742774685), (-0.013702414243129013, 0.10160028742774685), (-0.035967139113809665, 0.08889605405969991), (-0.047953015732060456, 0.07111214264811272), (-0.05163855865332412, 0.054698310911191594), (-0.050768254456645234, 0.04182611872848607)}, {(0.0608005750237699, 0.038537738209510675), (0.06584740126057076, 0.053661356222088674), (0.06665110726115575, 0.0760428541934095), (0.05539322350015552, 0.10533105377681062), (0.022868318524108144, 0.13045336248979789), (-0.022868318524107922, 0.13045336248979789), (-0.05539322350015552, 0.10533105377681107), (-0.06665110726115575, 0.0760428541934095), (-0.06584740126057076, 0.053661356222088674), (-0.0608005750237699, 0.038537738209510675)}, {(0.07120250793719496, 0.031600608004954456), (0.08266266010547563, 0.04716870945953655), (0.09364587512779754, 0.0748102843350471), (0.09221299546050243, 0.12277601478926714), (0.04524120071974669, 0.18070775060216882), (-0.0452412007197463, 0.18070775060216904), (-0.09221299546050243, 0.12277601478926736), (-0.09364587512779754, 0.07481028433504733), (-0.08266266010547563, 0.04716870945953677), (-0.07120250793719496, 0.03160060800495468)}, {(0.08007009503268847, 0.02032081957268983), (0.09903371862324573, 0.03231457377154601), (0.1269225345190832, 0.057980521450762934), (0.16185727131529482, 0.1232322406605082), (0.12337729774639195, 0.2818049641707342), (-0.12337729774639111, 0.2818049641707352), (-0.16185727131529504, 0.12323224066050875), (-0.12692253451908342, 0.057980521450763045), (-0.09903371862324573, 0.03231457377154601), (-0.08007009503268847, 0.02032081957268994)}, {(0.08488216962382156, 0.005412310058084541), (0.10884638667561086, 0.008923283323243991), (0.1514145406391274, 0.017378259777899724), (0.24665640085125506, 0.04718237970828895), (0.5770211525851817, 0.3311314568812653), (-0.5770211525851846, 0.33113145688127027), (-0.24665640085125595, 0.04718237970828926), (-0.15141454063912763, 0.01737825977789978), (-0.10884638667561086, 0.008923283323244019), (-0.08488216962382156, 0.0054123100580845684)}, {(0.08388987923372415, -0.010592156468904601), (0.10676430713610774, -0.017331868041575993), (0.1458742632612966, -0.03315324165029476), (0.22359967877936116, -0.08469684802248545), (0.33476104598737516, -0.38041027953110745), (-0.33476104598737483, -0.3804102795311108), (-0.22359967877936182, -0.08469684802248595), (-0.14587426326129682, -0.03315324165029487), (-0.10676430713610774, -0.01733186804157605), (-0.08388987923372415, -0.010592156468904601)}, {(0.07747666080809212, -0.02450492870255938), (0.09402852175879461, -0.03823724789054883), (0.1158571134609776, -0.06595956118630664), (0.13454514567230103, -0.12766499617769456), (0.08426219159718717, -0.2398599885806285), (-0.08426219159718651, -0.23985998858062907), (-0.13454514567230103, -0.127664996177695), (-0.11585711346097782, -0.06595956118630686), (-0.09402852175879461, -0.03823724789054883), (-0.07747666080809212, -0.02450492870255938)}, {(0.0678370438986633, -0.034346786115358574), (0.07696337113632934, -0.05010115556114969), (0.08380337527264303, -0.07637534882802255), (0.07731120144730741, -0.1174310294710994), (0.03524281349194591, -0.16059509329852562), (-0.03524281349194558, -0.16059509329852562), (-0.07731120144730719, -0.11743102947109962), (-0.08380337527264303, -0.07637534882802255), (-0.07696337113632978, -0.05010115556114969), (-0.0678370438986633, -0.034346786115358574)}, {(0.057396464792191626, -0.03996736153143132), (0.06082430982949649, -0.05445553063468722), (0.05966823856647552, -0.07478871265775311), (0.047666399401361303, -0.09957583056761665), (0.019044794543057242, -0.11935459307381846), (-0.019044794543057075, -0.11935459307381846), (-0.047666399401361303, -0.09957583056761665), (-0.05966823856647552, -0.07478871265775311), (-0.06082430982949649, -0.05445553063468722), (-0.057396464792191626, -0.03996736153143132)}, {(0.047729245010124544, -0.042305467168064936), (0.0476692729232111, -0.05432439868846428), (0.043268978619005694, -0.06903368861486836), (0.03167999999999993, -0.08423999999999987), (0.011866993774822865, -0.09466624579460925), (-0.011866993774822754, -0.09466624579460925), (-0.03167999999999993, -0.08423999999999987), (-0.043268978619005694, -0.06903368861486836), (-0.0476692729232111, -0.05432439868846428), (-0.047729245010124544, -0.042305467168064936)}}
Textfield(Pot) → InputBox1
Plane(xAxis, yAxis) → c