Cartesian coordinates are rectilinear two-dimensional or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The three axes of three-dimensional Cartesian coordinates, conventionally denoted the x-, y-, and z-axes (a notation due to Descartes) are chosen to be linear and mutually perpendicular. In three dimensions, the coordinates x, y, and z may lie anywhere in the interval
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRyYwx4hMHAyqyNDBGykVQdGyFWTTfy9kZwIHaPc7Nh8whcjt8fPdil50XP9aZdJeLlTN-NTDlndRze7btzchx2_HZ1gioP5i4xHs5dfppDYYA-9xluEWgTLpCh3iBbk0bQBJzbCrT78Qi/s400/20.gif)
The inversion of three-dimensional Cartesian coordinates is called 6-sphere coordinates.
The scale factors of Cartesian coordinates are all unity, h_i=1. The line element is given by
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEKSowet7gNrzcb3v2K3aj2bMCrIBK6gcYgVLtyQuiSua3DRepvr6zS7umB7pRKAQQOFXRGgK3ysqTvTAEzmJ1Wpnvgf0omhkfOPnVuNPV8hdtC90Gipuzr9umdvmkfASBHCrneW_8-8RR/s200/4.gif)
and the volume element by
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXr6LrCB9PVLtpApwJBX6mRxjAkMDkdpcgfSpmmHI4I_xmSWhGay9geEO82qF-KSKskAwo38ktlP4VDI29gRUnHTB_1Quqf7t4IC15zwTTL9VURieVPX7h8U9x0x9N3rgxERISY5knX61h/s200/5.gif)
The gradient has a particularly simple form,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5AMJgbbLgwRjGhHBC5Pi0o8RqnDjOM2BCZV3BdyEBDAFSxsYjsqYYjzjQBW7MRppeQxv5xfwOvcr4fy_9ESbUjoPAUXhzJzo6AZxGRwLh3DDnnsg6G5Ytxa-PqSlrZlB7dRKCRrtXqkZ3/s200/6.gif)
as does the Laplacian
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJ7fkTzz5j4_P6XZ6lnR028I7oknbpqmZS1AKqxnRHyyuUqT5FpeIwVQ4k3BawOvURf3CSzfyO8rONCbRWMqKMGuxl_PFMm_-Cz4LOR3_uIIhVA57dwuu0TMDEod9BYRaxkmMZ-sHgmtTe/s200/7.gif)
The vector Laplacian,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4cJeuqyJ9a9kGT04yRRq-kuRtblBFEHnTymhuFgw60NGnbGEg57V7BvhAOVPLRKmuShHEbjf8nW0mxLOlY9PfeNmvVXGU7bobEHInke1EGWw2G8N_SwvkqLAomVCqF3SEMyRjWNQXz6tI/s200/15.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_DH9JijyzU15Er5DT9Mb5ysI9jKwwxRCXnS_vcJmcAim95IcDidmKbyE0B593KfuDpIiR-oiPV3Yd9H72VULvvmS9KLUaimrpLP-IjE2C-n6EpSBur4z3iT3Q01BAHbohlROhoN7NjNBY/s400/22.gif)
The divergence is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4i4rSao1QwpBoK86qvpeJZXcu_mh66cz65ERj148qQSgzznVdn5maiGZq2bbHBHGTzF2c-SzXoxE1sPmE8puhsCxSZiuHnA2wKDQjNY5NHqOZpc_FIKgUy0Wrkf-saq4EwnsyXyDCG7Ke/s400/21.gif)
and the curl,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOb8S-HxVZbCRNMBosD0fOCGnGu-pU_Fn7Kx6JG3Sp1Fyojd6I9bP32BKDhMQWmVQf7eq3YzvcD71wREaZq5vrLaA6kMRh88-oGG2MoMj7U8fJEMFM2bKQZ8C93HiA1wJpozy9yggC86me/s400/19.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrYKEdcvg5mK-06wYlIRGGaJgL2iwoGn08OdAqe5VuxwuuG72w6MB7YhpvlT942M-RwTEycArVQ8cVrg_iOgQNUz-IuxUui5ic7eQJqOIYzmbBXZNH0NOvPVOGK5Z4iYWpukn4p7c_zvt7/s200/11.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrukrF3zYZI0I0dkoF-G7skLU8k4oNUi8kDjaZug3-58mS9V_rfglislYeAbWNa0ZcvRbWenBj42SY3mm5CnBEqls715c44POykzQRlpwaqgaFtWWa_O2TgiY6nzDZD82RC7PDx-nQ8_4y/s400/17.gif)
The gradient of the divergence,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhu0nBHlymIJ5WuX3i2FNM2gpjge8y71goF7RIwClpIU7CCr0DJxx-ftbbrWeTVu2rJWqCQ06cpO_Xysyr1iQclVEE2XJusJCbS9sujucTc945K22z-PgCjzHtb3dNdK3mAUN08Q6jMdRL4/s400/18.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYGeP6432psyRIAMJX4GXp7iqCS-IzYgCGU16bCoDo53Ht8-hX-xK-3mZSDJHPRDtGssAPJMHnZbMkoIsV7G1RZ42P5UVWd820R_QPvx5UohQRBzcNnsmro6o5i7Xu9GUKeKJB53DtIZrp/s200/13.gif)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhW2-5BNGzJCuEeqa7ni3ysaI_xwvapYUvSZz7NV1-oRj4GRATOP3GqqXx9r0pLH8b_Tcg2RVf-JfzitqN2u9XecfZDzWy1yt0Cb63OjisL_S2B6dZdx_2VIoPmi_ys-7_U39vxVE2uqPSi/s200/14.gif)
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