C.2 · 多变量微积分与向量分析
GRE Subject Test · GRE 数学 知识点 2 3:05 英语旁白 · 英中内嵌字幕
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This unit computes partial derivatives, gradients and directional derivatives; uses multiple integrals and coordinate changes; and uses derivative conditions to tell planes from curved surfaces.
本单元计算偏导数、梯度和方向导数；使用重积分和坐标变换；并用导数条件区分平面和曲面。
A partial derivative changes one coordinate and fixes the others.
偏导数只改变一个坐标、固定其他坐标。
If f is differentiable, the directional derivative along a unit vector u is the gradient dotted with u.
若 f 可微，沿单位向量 u 的方向导数等于梯度点乘 u。
Use a direction of length two without normalising, and the answer doubles.
若用长度为 2 的方向而不先单位化，答案就会翻倍。
Differentiability means a valid linear approximation, not merely partial derivatives existing at one point.
可微意味着存在有效的线性近似，而不只是某一点的偏导数存在。
Continuous first partials nearby are sufficient.
附近一阶偏导数连续是充分条件。
Then the chain rule for f of x of t, y of t is f x times x dash plus f y times y dash.
此时 f(x(t), y(t)) 的链式法则是 fₓx′ + f_y y′。
A multiple integral sums over a region, so describe the region before choosing limits.
重积分是对一个区域求和，所以先描述区域再选积分限。
In polar coordinates the area element is r d r d theta, not just d r d theta.
极坐标下面积元是 r dr dθ，而不只是 dr dθ。
A general change of coordinates uses the absolute value of the Jacobian: area scales by a non-negative factor.
一般的坐标变换用雅可比行列式的绝对值：面积按非负因子缩放。
If both first partials are constant on the whole plane, integrating gives a x plus b y plus c: a plane.
若两个一阶偏导数在整个平面上都是常数，积分得 ax + by + c：一个平面。
Parallel straight level lines are not enough, because e to the x has them and is curved.
平行的直线等高线不够，因为 e^x 有这样的等高线却是弯曲的。
Zero mixed partials are not enough either: x squared plus y squared.
混合偏导为零也不够：x² + y² 就是反例。
For f equals x squared plus 3 y squared, the gradient is 2x, 6y, so at 1, 1 the gradient is 2, 6.
f = x² + 3y² 的梯度是 (2x, 6y)，所以在 (1, 1) 处梯度是 (2, 6)。
Towards direction 3, 4, the unit vector is three fifths, four fifths.
朝方向 (3, 4)，单位向量是 (3/5, 4/5)。
The directional derivative is the length of the gradient's shadow on that line: 6.
方向导数就是梯度在这条直线上投影的长度：6。
A non-unit direction vector gives a scaled directional rate, not the derivative per unit distance.
非单位方向向量给出的是缩放后的变化率，而不是每单位距离的导数。
The polynomial is differentiable.
多项式可微。
Its gradient at 1, minus 1 is 1, minus 3; with the unit vector three fifths, four fifths, the directional derivative is minus 9 over 5.
它在 (1, −1) 处的梯度是 (1, −3)；与单位向量 (3/5, 4/5) 点乘，方向导数是 −9/5。
For the disk, x squared plus y squared is r squared and d A is r d r d theta, so the integral is pi over 2.
对单位圆盘，x² + y² = r²，dA = r dr dθ，所以积分是 π/2。
On each axis g is zero, so both partials at the origin are zero.
g 在两条坐标轴上都是零，所以原点处两个偏导数都是零。
It is continuous, because the absolute value of x y is at most half of x squared plus y squared.
它是连续的，因为 |xy| 不超过 (x² + y²)/2。
But along the diagonal, the remainder ratio is always a half, so it never tends to zero: g is not differentiable.
但沿对角线，余项比值恒为 1/2，不趋于零：g 不可微。
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。
A directional derivative needs a unit vector.
方向导数需要单位向量。
Existing partials do not give differentiability.
偏导数存在不保证可微。
A zero mixed partial does not force a plane.
混合偏导为零不迫使函数是平面。
And a coordinate change scales area by the absolute Jacobian.
坐标变换按雅可比行列式的绝对值缩放面积。