Suppose 𝑓 = 𝑓(𝑢, 𝑣, 𝑤), 𝑔 = 𝑔(𝑢, 𝑣, 𝑤), 𝑕 = 𝑕(𝑢. 𝑣. 𝑤), and 𝑢 = 𝑢(𝑟, 𝑠, 𝑡), 𝑣 =𝑣(𝑟, 𝑠, 𝑡), 𝑤 = 𝑤(𝑟, 𝑠, 𝑡). Then, write the formula for the Jacobian 𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡)

Given 𝑓 = 𝑓(𝑢, 𝑣, 𝑤), 𝑔 = 𝑔(𝑢, 𝑣, 𝑤), 𝑕 = 𝑕(𝑢. 𝑣. 𝑤), and 𝑢 = 𝑢(𝑟, 𝑠, 𝑡), 𝑣 =𝑣(𝑟, 𝑠, 𝑡), 𝑤 = 𝑤(𝑟, 𝑠, 𝑡), the formula for the Jacobian 𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) is as follows:

𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) = 𝜕(𝑓,𝑔,𝑕)𝜕(𝑢,𝑣,𝑤) * 𝜕(𝑢,𝑣,𝑤)𝜕(𝑟,𝑠,𝑡)

This can be further expanded as:

𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) = 𝜕𝑓𝜕𝑢 * 𝜕𝑢𝜕𝑟 + 𝜕𝑓𝜕𝑣 * 𝜕𝑣𝜕𝑟 + 𝜕𝑓𝜕𝑤 * 𝜕𝑤𝜕𝑟 + 𝜕𝑔𝜕𝑢 * 𝜕𝑢𝜕𝑠 + 𝜕𝑔𝜕𝑣 * 𝜕𝑣𝜕𝑠 + 𝜕𝑔𝜕𝑤 * 𝜕𝑤𝜕𝑠 + 𝜕𝑕𝜕𝑢 * 𝜕𝑢𝜕𝑡 + 𝜕𝑕𝜕𝑣 * 𝜕𝑣𝜕𝑡 + 𝜕𝑕𝜕𝑤 * 𝜕𝑤𝜕𝑡

This formula represents the partial derivatives of 𝑓, 𝑔, and 𝑕 with respect to 𝑟, 𝑠, and 𝑡, respectively.

Given 𝑓 = 𝑓(𝑢, 𝑣, 𝑤), 𝑔 = 𝑔(𝑢, 𝑣, 𝑤), 𝑕 = 𝑕(𝑢. 𝑣. 𝑤), and 𝑢 = 𝑢(𝑟, 𝑠, 𝑡), 𝑣 =𝑣(𝑟, 𝑠, 𝑡), 𝑤 = 𝑤(𝑟, 𝑠, 𝑡), the formula for the Jacobian 𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) is as follows:

𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) = 𝜕(𝑓,𝑔,𝑕)𝜕(𝑢,𝑣,𝑤) * 𝜕(𝑢,𝑣,𝑤)𝜕(𝑟,𝑠,𝑡)

This can be further expanded as:

𝜕(𝑓,𝑔,𝑕)𝜕(𝑟,𝑠,𝑡) = 𝜕𝑓𝜕𝑢 * 𝜕𝑢𝜕𝑟 + 𝜕𝑓𝜕𝑣 * 𝜕𝑣𝜕𝑟 + 𝜕𝑓𝜕𝑤 * 𝜕𝑤𝜕𝑟 + 𝜕𝑔𝜕𝑢 * 𝜕𝑢𝜕𝑠 + 𝜕𝑔𝜕𝑣 * 𝜕𝑣𝜕𝑠 + 𝜕𝑔𝜕𝑤 * 𝜕𝑤𝜕𝑠 + 𝜕𝑕𝜕𝑢 * 𝜕𝑢𝜕𝑡 + 𝜕𝑕𝜕𝑣 * 𝜕𝑣𝜕𝑡 + 𝜕𝑕𝜕𝑤 * 𝜕𝑤𝜕𝑡

where 𝜕𝑓𝜕𝑢, 𝜕𝑓𝜕𝑣, 𝜕𝑓𝜕𝑤, 𝜕𝑔𝜕𝑢, 𝜕𝑔𝜕𝑣, 𝜕𝑔𝜕𝑤, 𝜕𝑕𝜕𝑢, 𝜕𝑕𝜕𝑣, 𝜕𝑕𝜕𝑤 represent the partial derivatives of 𝑓, 𝑔, 𝑕 with respect to 𝑢, 𝑣, 𝑤 respectively, and 𝜕𝑢𝜕𝑟, 𝜕𝑢𝜕𝑠, 𝜕𝑢𝜕𝑡, 𝜕𝑣𝜕𝑟, 𝜕𝑣𝜕𝑠, 𝜕𝑣𝜕𝑡, 𝜕𝑤𝜕𝑟, 𝜕𝑤𝜕𝑠, 𝜕𝑤𝜕𝑡 represent the partial derivatives of 𝑢, 𝑣, 𝑤 with respect to 𝑟, 𝑠, 𝑡 respectively.