绿公式(Green's theorem)和斯托克斯公式(Stokes' theorem)是数学中常用的两个定理,它们在向量分析和微积分等领域具有重要的应用。本文将深入探讨这两个定理的原理和应用。 一、绿公式 绿公式是对曲面积分和曲线积分之间关系的一种描述。它的数学表达方式如下: 设平面区域D为闭合曲线C所围成的有向区域,f(x, y)...
斯托克斯公式(Stokes' theorem):是描述矢量场沿着曲线的环量和该矢量场通过曲面的通量之间的关系的定理。在三维空间中,斯托克斯公式的一种形式为: ∮C F·dr = ∬S curl(F)·n dS 其中,C是曲面S的边界曲线,F是一个有连续偏导数的矢量场,dr是沿着C的微小位移矢量,S是由C所围成的曲面,curl(F)是F的旋度...
流体力学:梯度算子积分,Gauss定理,Stokes定理,Green定理。 Stokes定理证明,可参考:https://math.libretexts.org/Courses/Montana_State_University/M273%3A_Multivariable_Calculus/16%3A_Vector_Fields_Line_Integrals_and_Vector_Theorems/Stokes'_Theorem%20。 (视频里,“Stokes”,可能全都说成了“stocks”。抱歉。
Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem 340 Now let’s begin. Suppose the curve below is oriented in the counterclockwise direction and is parametrized by x. Suppose also that the top part of our curve corresponds to the function ...
The Fundamental Theorem of Calculus states that differentiation and integration are inverse processes. An appropriate extension of this theorem to double integrals of functions of two variables is known as Green's Theorem. Suppose that P and Q are smooth (i.e., continuously differentiable) functions...
基于以上讨论,我们发现斯托克斯定理(Stokes’ Theorem)可以用微分形式(differential forms)重新表述。 考虑1 形式(1-form) \alpha = {\alpha }_{x}{dx} + {\alpha }_{y}{dy} + {\alpha }_{z}{dz} 及其对应的矢量场 F = {\alpha }_{x}\widehat{i} + {\alpha }_{y}\widehat{j} + {\alp...
Problem 3: Verify Stokes' theorem for the case in which is the portion of the upper sheet of the hyperbolic paraboloid 1 2 2 2 y x z that lies below the plane 5 z , and F5 is as the following input cell. F5=[-z*y,z*x,x^2+y^2] More on Green's Theorem Let's go...
The usefulness of this method and operative indications for craniopharyngioma are discussed. It may be concluded that this method is less invasive and safer for removal of the tumor, with preservation or even the possibility of improvement in the pituitary function, when the tumor is mainly located...
I=∮C(5y+x)dx+(y+4x)dy Green's Theorem: Green's Theorem lets us transform closed line integrals into area integrals. It is a lower dimensional version of Stokes' Theorem (the special case where thez-component is 0). The theorem is: ...