< p > TL;博士:这取决于如何定义土地面积。与一个合理的定义,这样的子午线(连同其完成地球的另一边)一定会存在由于中间值定理。也是一样的自由。< / p > <人力资源> < p > < / p > < ul >定义函数<李> < p > <跨类= " math-container " > \美元operatorname{一}_ {E}}{\文本(\λ)< / span >美元的面积是陆地的半球投射到参考椭球体的东部的经度<跨类= " math-container " > \λ< / span >美元。李李< / p > < / > < > < p > <跨类= " math-container " > \美元operatorname{一}_ {W}}{\文本(\λ)$ < / span >土地的面积是投射到参考椭球体的半球向西经度线类< span = " math-container " > \λ< / span >美元。李李< / p > < / > < > < p > <跨类= " math-container " > \美元operatorname{\δA} _{朗}}{\文本(\λ)$ < / span >的区别是< span class = " math-container " > $ \ operatorname{一}_ {E}}{\文本(\λ)$ < / span >和<跨类= " math-container " > $ \ operatorname{一}_ {W}}{\文本(\λ)$ < / span >。李李< / p > < / > < > < p > <跨类= " math-container " > \美元operatorname{一}_ {N}}{\文本(\φ)$ < / span >土地的面积是投射到参考椭球体半球向北的纬度线类< span = " math-container " > \φ< / span >美元。李李< / p > < / > < > < p > <跨类= " math-container " > \美元operatorname{一}_{年代}}{\文本(\φ)$ < / span >土地的面积是投射到参考椭球体的半球向西经度线类< span = " math-container " > \φ< / span >美元。李李< / p > < / > < > < p > <跨类= " math-container " > \美元operatorname{\δA} _ {lat}}{\文本(\φ)$ < / span >的区别是< span class = " math-container " > $ \ operatorname{一}_ {N}}{\文本(\φ)$ < / span >和<跨类= " math-container " > $ \ operatorname{一}_{年代}}{\文本(\φ)$ < / span >。< / p > < /李> < / ul > < p >我的定义让每一个连续函数。注意:其他土地面积可能导致非连续函数的定义。想象一个完全垂直的悬崖,从北到南。如果这个悬崖的面积计算土地面积那么<跨类= " math-container " > \美元operatorname{\δA} _{\文本{朗}}(\λ)< / span >美元不会是连续的。这同样适用于< span class = " math-container " > $ \ operatorname{\δA} _{\文本{lat}}(\φ)< / span >美元完全垂直的悬崖,东到西。< / p > < p >证明纬度线必须存在是很容易的,所以我先这样做。 All of the Earth's land area is north of 90° south latitude, making $\operatorname{\Delta A}_{\text{lat}}(-90)$ a large positive number. All of the Earth's land area is south of 90° north latitude, making $\operatorname{\Delta A}_{\text{lat}}(90)$ a large negative number. Because zero is between this large negative number and large positive number, and because $\operatorname{\Delta A}_{\text{lat}}(\phi)$ is continuous, there must necessarily exist at least one line of latitude $\phi$ for which $\operatorname{\Delta A}_{\text{lat}}(\phi)$ is zero. Regarding longitude, Pick an arbitrary longitude $\lambda$. If $\operatorname{\Delta A}_{\text{lon}}(\lambda)$ is zero we have a winner. If it's not zero, then since $\operatorname{\Delta A}_{\text{lon}}(\lambda +180°) = -\operatorname{\Delta A}_{\text{lon}}(\lambda)$, there exists at least one longitude $\lambda_0$ between $\lambda$ and $\lambda+180°$ where $\operatorname{\Delta A}_{\text{lon}}(\lambda_0)$ is zero.