Thay đổi tọa độ và các định lý tích phân vector
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| curl/kɜːl/ | A vector measuring the local rotation of a field. |
| divergence/daɪˈvɜːdʒəns/ | divergence |
| simply connected/ˈsɪmpli kəˈnektɪd/ | Connected, with every closed loop shrinkable to a point inside the region. |
Zero curl 旋度, nonzero circulation
- $\mathbf F = \left(\dfrac{-y}{x^2+y^2}, \dfrac{x}{x^2+y^2}\right)$ has zero curl wherever it is defined, yet its circulation around the unit circle is $2\pi$.
- The hole at the origin is the reason. GRE vector calculus tests Jacobians, Green, Stokes, divergence 散度 and conservative fields — with their hypotheses.
Coordinate changes
- Polar $r\,dr\,d\theta$; cylindrical $r\,dr\,d\theta\,dz$; spherical ($\varphi$ from the $+z$-axis) $\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$.
- Transform the integrand and the region; the Jacobian enters as an absolute value.
The spherical volume element ($\varphi$ measured from the $z$-axis) is
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The Jacobian is ρ² sin φ.
Green's theorem
- $\oint_C P\,dx + Q\,dy = \iint_R (Q_x - P_y)\,dA$, with $C$ counter-clockwise and $P, Q$ continuously differentiable on a region containing $R$.
- $(-y, x)$ around the unit circle: $\iint 2\,dA = 2\pi$. Clockwise gives $-2\pi$.
The counter-clockwise circulation of $(-y, x)$ around the circle of radius 3 is $k\pi$. Find $k$.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
By Green's theorem the circulation is ∬ (∂Q/∂x − ∂P/∂y) dA = ∬ (1 − (−1)) dA = 2 × area = 2 · 9π = 18π, so k = 18.
Stokes and divergence
- Stokes: boundary circulation = flux of $\nabla \times \mathbf F$ through the surface, orientation by the right-hand rule.
- Divergence: outward flux through a closed surface = $\iiint \nabla \cdot \mathbf F\,dV$.
$\mathbf F = (x, y, z)$: $\nabla \cdot \mathbf F = 3$. Flux out of the sphere of radius 2 is $3 \cdot \tfrac43\pi \cdot 8 = 32\pi$ — no surface parameterisation needed.
Which theorem?
Match an integral to the theorem that converts it.
The outward flux of $\mathbf F = (x, y, z)$ through the unit sphere is $k\pi$. Find $k$.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
By the divergence theorem, flux = ∭ div F dV = ∭ 3 dV = 3 · (4/3)π · 1³ = 4π, so k = 4.
Conservative fields
- Gradient fields have path-independent line integrals: $\int_C \nabla\varphi \cdot d\mathbf r = \varphi(B) - \varphi(A)$.
- Curl-free on a simply connected 单连通 domain ⇒ conservative. With a hole, curl-free is not enough.
A curl-free field on $\mathbb R^2 \setminus \{0\}$ is conservative.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The angular field circulates 2π.
$\mathbf F = \nabla(x^2 y)$. Find $\int_C \mathbf F \cdot d\mathbf r$ along any path from $(0,0)$ to $(2, 3)$.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$\varphi(2,3) - \varphi(0,0) = 12$.
Recap
- Include $|\det J|$: $r$, $r$, $\rho^2\sin\varphi$.
- Green for plane circulation, Stokes for surface boundaries, divergence for closed-surface flux.
- Curl-free implies conservative only on simply connected domains.
- curl: A vector measuring the local rotation of a field.
- simply connected: Connected, with every closed loop shrinkable to a point inside the region.