Scribd
Upload
16 views2 pages

Summary 1

Uploaded by

lmoum861
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
16 views2 pages

Summary 1

Uploaded by

lmoum861
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

Recap of coordinates Cartesian coordinate Polar coordinate Cylindrical coordinate Spherical coordinate

𝒓
𝒓
𝒓
𝜌= 𝑥 + 𝑦 + 𝑧²
𝑟= 𝑥 + 𝑦² 𝜌= 𝑥 + 𝑦²
𝑦
𝑦 𝑦 𝜑 = tan( )
𝜑 = tan ( ) 𝑥
𝜃 = tan ( ) 𝑥
𝑥 𝑧≡𝑧 𝑥 +𝑦
𝜃 = tan ( )
𝑧
𝑢 = cos 𝜃 𝚤⃗ + sin 𝜃 𝚥⃗ 𝑢 = cos 𝜑 𝚤⃗ + sin 𝜑 𝚥⃗ 𝒖𝒓 = 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝝋 ⃗ + 𝐬𝐢𝐧 𝜽 𝐬𝐢𝐧 𝝋 ⃗ + 𝐜𝐨𝐬 𝜽 𝒌
𝑢 = − sin 𝜃 𝚤⃗ + cos 𝜃 𝚥⃗ 𝑢 = − sin 𝜑 𝚤⃗ + cos 𝜑 𝚥⃗ 𝒖𝜽 = 𝐜𝐨𝐬 𝜽 𝐜𝐨𝐬 𝝋 ⃗ + 𝐜𝐨𝐬 𝜽 𝐬𝐢𝐧 𝝋 ⃗ − 𝐬𝐢𝐧 𝜽 𝒌
𝒖𝝋 = − 𝐬𝐢𝐧 𝝋 ⃗ + 𝐜𝐨𝐬 𝝋 ⃗
𝑘≡𝑘

d 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 + 𝒅𝒛𝒌 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 + 𝒓 sin 𝜽 𝒖𝝋

𝑑𝑆 = 𝜌𝑑𝜌𝑑𝜑
𝑑𝑆 = 𝑑𝜌𝑑𝑧
𝑑𝑆 = 𝜌𝑑𝜑𝑑𝑧

𝝏 𝟏 𝝏 𝝏 𝟏 𝝏 𝝏 𝝏 𝟏 𝝏 𝟏 𝝏
𝜵= 𝒖𝒓 + 𝒖𝜽 𝜵 = 𝒖 𝝆 + 𝒖 𝝋 + 𝒌 𝜵= 𝒖𝒓 + 𝒖𝜽 + 𝒖
𝝏𝝆 𝝆 𝝏𝝋 𝝏𝒛 𝝏𝒓 𝒓 𝝏𝜽 𝒓 𝒔𝒊𝒏 𝜽 𝝏𝝋 𝝋
𝝏𝒓 𝒓 𝝏𝜽
Position vector Cartesian coordinate Polar coordinate Cylindrical coordinate Spherical coordinate

𝒓
𝒓
𝒓
Elementary displacement

d 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 + 𝒅𝒛𝒌 𝒅𝒓 𝒕 = 𝒅𝒓𝒖𝒓 + 𝒓𝒅𝜽𝒖𝜽 + 𝒓 sin 𝜽 𝒖𝝋


Elementary volume
Nabla Elementary Surface

𝑑𝑆 = 𝜌𝑑𝜌𝑑𝜑
𝑑𝑆 = 𝑑𝑥𝑑𝑦
𝑑𝑆 = 𝑑𝜌𝑑𝑧
𝑑𝑆 = 𝑑𝑥𝑑𝑧
𝑑𝑆 = 𝑑𝑦𝑑𝑧 𝑑𝑆 = 𝜌𝑑𝜑𝑑𝑧

𝝏 𝟏 𝝏 𝝏 𝟏 𝝏 𝝏 𝝏 𝟏 𝝏 𝟏 𝝏
𝜵 = 𝒖 + 𝒖 + 𝒌 𝜵= 𝒖𝒓 + 𝒖𝜽 + 𝒖
𝜵= 𝒖𝒓 + 𝒖𝜽 𝝏𝝆 𝝆
𝝆 𝝏𝝋 𝝋
𝝏𝒛 𝝏𝒓 𝒓 𝝏𝜽 𝒓 𝒔𝒊𝒏 𝜽 𝝏𝝋 𝝋
𝝏𝒓 𝒓 𝝏𝜽

You might also like