In this video I derive the Laplace Operator or Laplacian in spherical coordinates by applying the Laplacian in polar coordinates twice: once for the azimuthal (ф) angle and once for the polar angle (θ). Adding the two polar Laplacians, cancelling out terms, and writing the partial derivatives in terms of the spherical coordinates, we obtain the Laplace Operator in spherical coordinates!
#math #polarcoordinates #calculus #sphericalcoordinates #multivariablecalculus
Timestamps
- Converting rectangular coordinates to spherical coordinates – 0:00
- Will apply the Laplacian in polar coordinates twice – 9:25
- Applying the Laplacian in polar coordinates to the x-y terms – 10:38
- Applying the Laplacian in polar coordinates to the z-s terms – 13:43
- Adding the two Laplacians and canceling out terms – 17:15
- Evaluating the partial derivative in terms of s – 20:20
- Combining all the terms, we obtain our Laplace Operator in spherical coordinates – 30:31
- Simplifying the Laplacian to match up with Wikipedia – 37:14
- Double-checking our Laplacian in Spherical Coordinates with Wikipedia – 39:57 .
Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
MES Links: https://mes.fm/links
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: @mes
Email me: [email protected]
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
▶️ 3Speak