最近30从www.hoelymoley.com 2023 - 08 - 28 - t20:37:47z //www.hoelymoley.com/feeds/question/23480 https://creativecommons.org/licenses/by-sa/4.0/rdf //www.hoelymoley.com/q/23480 3 拉法 //www.hoelymoley.com/users/26265 2022 - 01 - 31 - t10:28:37z 2022 - 02 - 01 - t08:23:14z

< p >我目前使用GGM05C斯托克斯系数重建地球的引力势的梯度。< / p > < p >我发现在球坐标的表达式表示梯度< a href = " http://icgem.gfz potsdam.de/str - 0902 revised.pdf”rel = " nofollow noreferrer " >这一技术报告ICGEM < / >。特别是,方程122页23显示了重力势的偏导数<跨类= " math-container " > W < / span >对美元三个球坐标参数<跨类= " math-container " > r < / span >美元(距离中心),< span class = " math-container " > \λ< / span >美元(经度)和<跨类= " math-container " > \ varphi < / span >美元(地心纬度)。< / p > < p >这些方程涉及相关的勒让德函数的使用<跨类= " math-container " >美元P_ {lm} $ < / span >,这是一个纬度的函数类< span = " math-container " > \ varphi < / span >美元。我理解,当执行偏导数对< span class = " math-container " > $ r $ < / span >, < span class = " math-container " > \ varphi美元< / span >和<跨类= " math-container " > \λ< / span >美元,勒让德函数保持不变的衍生品对< span class = " math-container " > $ r $ < / span >和<跨类= " math-container " > \λ< / span >,美元自勒让德函数不是一个函数的< span class = " math-container " > $ r < / span >或<跨类= " math-container " > \λ< / span >美元。然而,由于他们是一个纬度的函数类< span = " math-container " > \ varphi < / span >美元,当我们计算类< span = " math-container " > \美元dfrac{\部分W}{\部分\ varphi} $ < / span >,我们需要得到相关的勒让德函数,获得以下表达式,表示122年方程的相关文档:< / p > < p > <跨类= " math-container " > $ $ \ dfrac{\部分W}{\部分\ varphi} = \压裂{通用}{r} \ sum_ {\ mathscr {l} = 0} ^ {\ mathscr {l} _{马克斯}}\离开(\压裂{r} {r} \右)^ \ mathscr {l} \ sum_ {m = 0} ^ {\ mathscr {l}} \ dfrac{\部分P_ {\ mathscr {l} m}(罪\ \ varphi)}{\部分\ varphi} \离开(C_ {\ mathscr {l} m} ^海关组织(m \λ)+ S_ {\ mathscr {l} m} ^ Wsin (m \λ)\右)$ $ < / span > < / p > < p >然而,链式法则后,不能这衍生品还包括一个乘法<跨类= " math-container " >因为\ \ varphi < / span >美元,因为这是的导数< span class = " math-container " >罪\ \ varphi < / span >美元嵌套在< span class = " math-container " >美元P_ {\ mathscr {l} m}(罪\ \ varphi) < / span >美元? < / p >

//www.hoelymoley.com/questions/23480/-/23486 # 23486 3 Joscha Fregin //www.hoelymoley.com/users/18833 2022 - 02 - 01 - t08:23:14z 2022 - 02 - 01 - t08:23:14z

< p >链式法则已经包括在内,因为导数是采取对< span class = " math-container " > \ varphi < / span >美元。注意,< span class = " math-container " > \开始{方程}\压裂{\部分P_{\魔法m}(\罪(\ varphi)}{\部分\ varphi} = \压裂{\部分P_{\魔法m}(\罪(\ varphi)}{\部分\罪(\ varphi)} \压裂{\部分\罪(\ varphi)}{\部分\ varphi}。结束\{方程}< / span > < / p >

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