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Review of mathematical knowledge: triple integral

2022-06-22 02:09:00 【UQI-LIUWJ】

1 Triple integrals in rectangular coordinates

Calculation method of object volume $\iiint_\Omega 1\cdot dv$

1.1 First definite integral, then double integral

set up D by Oxy The area on the plane ,C Is its boundary .

If the space area Ω By Region D The surface on $z=z_1(x,y),z=z_2(x,y)\quad [z_1(x,y) \le z_2(x,y)]$ as well as With C Guide line & a bus or bus bar , Parallel to the x Surrounded by the cylindrical surface of the shaft
function f(x,y,z) stay Ω Continuous on

be $\iiint_D ff(x,y,z)dz=\iint_D[\int_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz]dxdy$

1.1.1 give an example

Calculation $I=\iiint_\Omega(x^2+y^2)dv$ ,Ω By Enclosed area .

$I=\iint_D[\int_{\frac{x^2+y^2}{2}}^2(x^2+y^2)dz]d\sigma$
$=\iint_D[(2-\frac{x^2+y^2}{2})(x^2+y^2)]d\sigma$
Use polar transformation , Yes ：

$=\int_0^{2\pi}[\int_0^2(2-\frac{r^2}{2})\cdot r^2 \cdot rdr]d\theta$
$=\int_0^{2\pi}[\int_0^2 (2r^3-\frac{1}{2}r^5) rdr]d\theta$
$=\frac{16}{3}\pi$

1.2 Double integral first, then definite integral

If the space area Ω Between planes z=a Peace surface z=b (a<b) Between , Plane $z=z_0,(z_0 \in [a,b])$ And Ω Intersect on a plane $\Omega(z_0)$ , be $\iiint_\Omega f(x,y,z)dv=\int_a^b [\iint_{\Omega(z)}f(x,y,z)dxdy]dz$

1.2.1 1.1.1 The second solution of

$I=\int_0^2[\iint_\Omega (x^2+y^2)d\sigma]dz$

$=\int_0^2\{\int_0^{2\pi}[\int_0^{\sqrt{2z}} (r^2)rdr]d\theta\}dz =\frac{16}{3}\pi$

2 Cylindrical coordinate system

set up M For a point in space , It's in Oxy Projection point on plane P The polar coordinates of are (r,θ), its z The axis coordinates are z.

We call (r,θ,z) by M The cylindrical coordinates of the point .

Rectangular coordinate system and coordinate system (r,θ,z) The relationship between is $\left\{\begin{matrix} x=rcos\theta\\ y=rsin\theta\\ z=z \end{matrix}\right.$

Coordinate transformation $\iiint_\Omega f(x,y,z)dv=\iiint_\Omega' f(rcos\theta,rsin\theta,z)rdrd\theta dz$ among $\Omega'=\{(r,\theta,z)|(rcos\theta,rsin\theta,z) \in \Omega\}$

3 Spherical coordinate system

set up M For a point in space ,ρ Indication point M The distance to the origin ,θ Indicates an over point M And z The half plane of the axis and Oxz The angle between the planes ,φ Express z Axes and vectors OM The angle between .

$(\rho,\theta,\varphi)$ yes M Spherical coordinates of $(\rho \in [0,+\infty], \theta \in [0,2\pi),\varphi \in [0,\pi])$

So rectangular coordinates (x,y,z) And spherical coordinates $(\rho,\theta,\varphi)$ The relationship is ： $\begin{matrix} x=\rho sin\varphi cos\theta\\ y=\rho sin\varphi sin\theta\\ z=\rho cos\varphi \end{matrix}$

Use spherical coordinates to calculate triple integral , Yes $\iiint_\Omega f(x,y,z)dv=\iiint_{\Omega'} f(\rho sin\varphi cos\theta,\rho sin\varphi sin\theta,\rho cor \varphi) \rho^2 sin\varphi d\rho d\theta d\varphi$

4 Commutative integral

Similar to double integral

Let's set the function f(x,y,z) In a bounded closed region Ω Continuous on , If the variable replaces $\left\{\begin{matrix} x=x(u,v,w)\\ y=y(u,v,w)\\ z=z(u,v,w) \end{matrix}\right.$ Three conditions are satisfied ：

take uvw Areas in space Ω' One by one Ω
Transform function x(u,v,w),y(u,v,w),z(u,v,w) stay Ω' Continuous on , And have a continuous first-order partial derivative
Jacobian determinant $J(u,v,w)=\frac{D(x,y,z)}{D(u,v,w)}$ stay Ω' Do not take zero value on