当前位置：网站首页>Review of mathematical knowledge: curve integral of type I

Review of mathematical knowledge: curve integral of type I

2022-06-29 16:34:00 【UQI-LIUWJ】

1 The formula

If the curve C The parametric equation of is $\left\{\begin{matrix} x=\varphi(t)\\ y=\phi(t) \end{matrix}\right.$ , $\varphi '(t),\phi '(t)$ stay [α,β] Continuous on ,f(x,y) stay C Continuous on , So the curve integral of the first type $\int_C f(x,y)ds$ It can be converted into definite integral
$\int_c f(x,y)ds=\int_\alpha^\beta f(\varphi(t),\phi(t))\sqrt{[\varphi' (t)]^2+[\phi'(t)^2]}dt$

A special case ： curve C The equation of y=g(x), So the curve integral of the first type $\int_C f(x,y)ds$ It can be converted into definite integral $\int_c f(x,y)ds=\int_\alpha^\beta f(x,g(x))\sqrt{1+[g'(x)]^2}dx$

1.1 The space curve C First type curve integral of

If the space curve C The parameter equation of is $x=\varphi(t),y=\phi(t),z=\chi(t), [\alpha \le t \le \beta]$ , $\varphi'(t),\phi'(t),\chi'(t)$ stay $[\alpha,\beta]$ Continuous on ,f(x,y,z) stay C Continuous on , So there are ：