Worksheet: Laplacian of a Function

In this worksheet, we will practice finding the Laplacian of a scalar function in Cartesian, cylindrical, and spherical coordinates.

Q1:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘₯+𝑦+𝑧 in Cartesian coordinates.

Q2:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘₯+𝑦+π‘§οŠ©οŠ©οŠ© in Cartesian coordinates.

  • A18(π‘₯+𝑦+𝑧)
  • B3ο€Ήπ‘₯+𝑦+π‘§ο…οŠ¨οŠ¨οŠ¨
  • C6(π‘₯+𝑦+𝑧)
  • D(π‘₯+𝑦+𝑧)
  • Eο€Ήπ‘₯+𝑦+π‘§ο…οŠ¨οŠ¨οŠ¨

Q3:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘’οŠ±ο—οŠ±ο˜οŠ±ο™οŽ‘οŽ‘οŽ‘ in Cartesian coordinates.

  • A𝑒4π‘₯+4𝑦+4π‘§ο…οŠ±ο—οŠ±ο˜οŠ±ο™οŠ¨οŠ¨οŠ¨οŽ‘οŽ‘οŽ‘
  • Bβˆ’2π‘₯π‘’βˆ’2π‘¦π‘’βˆ’2π‘§π‘’οŠ±ο—οŠ±ο˜οŠ±ο™οŠ±ο—οŠ±ο˜οŠ±ο™οŠ±ο—οŠ±ο˜οŠ±ο™οŽ‘οŽ‘οŽ‘οŽ‘οŽ‘οŽ‘οŽ‘οŽ‘οŽ‘
  • C𝑒4π‘₯+4𝑦+4π‘§βˆ’6ο…οŠ±ο—οŠ±ο˜οŠ±ο™οŠ¨οŠ¨οŠ¨οŽ‘οŽ‘οŽ‘
  • D𝑒4π‘₯+4𝑦+4π‘§ο…ο—οŠ°ο˜οŠ°ο™οŠ¨οŠ¨οŠ¨οŽ‘οŽ‘οŽ‘
  • Eπ‘’ο€Ήβˆ’4π‘₯βˆ’4π‘¦βˆ’4π‘§ο…οŠ±ο—οŠ±ο˜οŠ±ο™οŠ¨οŠ¨οŠ¨οŽ‘οŽ‘οŽ‘

Q4:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=ο€Ήπ‘₯+𝑦+π‘§ο…οŠ¨οŠ¨οŠ¨οŽ’οŽ‘ in terms of the spherical coordinates 𝜌, πœƒ, and πœ™.

  • A12
  • B6
  • C6𝜌
  • D12𝜌
  • E12𝜌

Q5:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘₯.

  • Aπ‘₯οŠͺ
  • B5π‘₯οŠͺ
  • C20π‘₯οŠͺ
  • D20π‘₯
  • E5π‘₯

Q6:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘’οŠ±ο—οŠ±ο˜οŠ±ο™οŽ‘οŽ‘οŽ‘ in terms of the spherical coordinates 𝜌, πœƒ, and πœ™.

  • A4ο€½1βˆ’1πœŒο‰π‘’οŠ¨οŠ±οŽ…οŽ‘
  • Bο€Ή4𝜌+6ο…π‘’οŠ¨οŠ±οŽ…οŽ‘
  • C2ο€½1βˆ’1πœŒο‰π‘’οŠ¨οŠ°οŽ…οŽ‘
  • Dο€Ή4πœŒβˆ’6ο…π‘’οŠ¨οŠ±οŽ…οŽ‘
  • E4ο€½1+1πœŒο‰π‘’οŠ¨οŠ±οŽ…οŽ‘

Q7:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=ο€Ήπ‘₯+𝑦+π‘§ο…οŠ¨οŠ¨οŠ¨οŽ’οŽ‘.

  • A3π‘₯√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨
  • B6π‘₯√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨
  • C12√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨
  • Dπ‘₯√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨
  • E2π‘₯√π‘₯+𝑦+π‘§οŠ¨οŠ¨οŠ¨

Q8:

Find the Laplacian of the function 𝑓(π‘₯,𝑦,𝑧)=π‘’ο—οŠ°ο˜οŠ°ο™ in Cartesian coordinates.

  • Aπ‘’ο—οŠ°ο˜οŠ°ο™
  • B3π‘’ο—οŠ°ο˜οŠ°ο™
  • C16π‘’ο—οŠ°ο˜οŠ°ο™
  • D6π‘’ο—οŠ°ο˜οŠ°ο™
  • E13π‘’ο—οŠ°ο˜οŠ°ο™

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.