4种积分

$$\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx.$$

$$\iint_{D} f(x,y) \, d\sigma = \int_{a}^{b} \int_{c}^{d} f(x,y) \, dx \, dy.$$

$$\iiint_{\Omega} f(x,y,z) \, dV = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) \, dx \, dy \, dz.$$

$$\int_{C} f(x,y) \, ds = \int_{a}^{b} f(x(t),y(t)) \sqrt{x'(t)^{2} + y'(t)^{2}} \, dt.$$

一型曲线积分是对曲线弧长的积分.

$$\int_{C} P(x,y) \, dx + Q(x,y) \, dy = \int_{a}^{b} P(x(t),y(t)) x'(t) \, dt + Q(x(t),y(t)) y'(t) \, dt.$$

二型曲线积分是对坐标的积分.

$$\iint_{\Sigma} f(x,y,z) \, dS = \iint_{D} f(x(u,v),y(u,v),z(u,v)) \lvert \mathbf{r}_{u} \times \mathbf{r}_{v} \rvert \, du \, dv.$$

第一型曲面积分是对曲面面积的积分.

$$\begin{align}& \iint_{\Sigma} P(x,y,z) \, dy \, dz + Q(x,y,z) \, dz \, dx + R(x,y,z) \, dx \, dy \\\\&= \iint_{D} P(x(u,v),y(u,v),z(u,v)) \lvert \mathbf{r}_{u} \times \mathbf{r}_{v} \rvert \, dv \, du. \end{align}$$

第二型曲面积分是对坐标的积分.

$$\int_{C} P(x,y) \, dx + Q(x,y) \, dy = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dx \, dy.$$

Green公式的几何意义: 曲线积分等于曲线围成的区域的面积.

$$\begin{align}& \iint_{\Sigma} P(x,y,z) \, dy \, dz + Q(x,y,z) \, dz \, dx + R(x,y,z) \, dx \, dy \\\\&= \iiint_{\Omega} \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \, dx \, dy \, dz. \end{align}$$

Gauss公式的几何意义: 曲面积分等于曲面围成的区域的体积.

$$\begin{align}& \int_{C} P \, dx + Q \, dy + R \, dz \\\\&= \iint_{S} \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \, dy \, dz + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \, dz \, dx + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dx \, dy.\end{align}$$

Stokes公式的几何意义: 曲线积分等于曲线围成的曲面的面积.