ج.7 · تغيير الإحداثيات ونظريات التكامل الشعاعي
GRE Subject Test · GRE Mathematics الموضوع 18 2:51 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
فصول
النص الصوتي
This unit transforms double and triple integrals with their Jacobians, applies the Green, Stokes and divergence theorems with correct orientation, and checks domain singularities before asserting path independence.
本单元用雅可比行列式变换二重和三重积分，以正确的定向应用格林、斯托克斯和散度定理，并在断言路径无关之前检查定义域中的奇点。
Changing variables scales area or volume by the absolute Jacobian determinant: r d r d theta in polar, an extra d z in cylindrical, and rho squared sine phi in spherical, with phi measured from the positive z-axis.
换元时面积或体积按雅可比行列式的绝对值缩放：极坐标是 r dr dθ，柱坐标多一个 dz，球坐标是 ρ² sin φ，其中 φ 从正 z 轴量起。
A missing Jacobian gives the wrong quantity.
漏掉雅可比因子就会算出错误的量。
Green's theorem equates positively oriented boundary circulation with the double integral of Q x minus P y; a hole needs its own inner boundary.
格林定理把正向边界环量与 Q_x − P_y 的二重积分等同；有洞时需要它自己的内边界。
Stokes links circulation to curl through the right-hand rule.
斯托克斯定理通过右手法则把环量与旋度联系起来。
The divergence theorem equates outward flux across a closed surface with the integral of div F.
散度定理把穿过闭曲面的向外通量与 div F 的积分等同。
A gradient field has path-independent line integrals.
梯度场的线积分与路径无关。
A smooth curl-free field on a simply connected domain is conservative, but curl-free alone on a domain with a hole is not enough.
单连通区域上光滑的无旋场是保守场，但在有洞的区域上仅无旋是不够的。
The field minus y, x, over x squared plus y squared, excludes the origin; its unit-circle circulation is 2 pi.
场 (−y, x)/(x² + y²) 不包括原点；它沿单位圆的环量是 2π。
For F equals x, y, z, the divergence is 3, so the outward flux through the sphere of radius 2 is 3 times its volume: 32 pi, with no surface parametrisation.
对 F = (x, y, z)，散度为 3，所以穿过半径为 2 的球面的向外通量是其体积的 3 倍：32π，无需参数化曲面。
For the planar field minus y, x, Green gives 2 integrated over the unit disk: 2 pi counterclockwise.
对平面场 (−y, x)，格林定理给出 2 在单位圆盘上的积分：逆时针为 2π。
Zero curl is not enough when the domain has a hole.
区域有洞时，旋度为零是不够的。
Outward flux and counterclockwise circulation use different theorems and orientation rules.
向外通量和逆时针环量使用不同的定理和定向规则。
In polar coordinates the annulus has r from 1 to 2.
在极坐标下，圆环的 r 从 1 到 2。
Including the Jacobian, the integrand becomes r cubed, and the integral is 15 pi over 2.
加上雅可比因子后被积函数变为 r³，积分为 15π/2。
The field x, y, z has divergence three on the whole ball, so the outward flux across the unit sphere is 4 pi.
场 (x, y, z) 在整个球内散度为 3，所以穿过单位球面的向外通量是 4π。
On the counterclockwise unit circle, G equals the velocity of the parametrisation, so the integrand is 1 and the circulation is 2 pi.
在逆时针单位圆上，G 等于参数化的速度，所以被积函数为 1，环量是 2π。
The arrows circulate around the hole.
箭头绕着洞旋转。
Green's theorem does not apply, because G is undefined at the origin, inside the disk.
格林定理不适用，因为 G 在圆盘内的原点处没有定义。
Check yourself.
自我检测。
The polar area element is r d r d theta.
极坐标面积元是 r dr dθ。
The flux of x, y, z through the sphere of radius 2 is 32 pi.
(x, y, z) 穿过半径 2 球面的通量是 32π。
Curl-free on a punctured plane need not be conservative.
挖去一点的平面上无旋场不一定是保守场。
Reversing orientation flips the sign.
反转定向改变符号。
And the divergence theorem needs a closed surface.
散度定理需要闭曲面。