C.7 · 좌표 변환 및 벡터 적분 정리
GRE Subject Test · GRE Mathematics 주제 18 2:51 영어 내레이션 · 영어 + 중국어 자막 burned-in
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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。