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 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 in Cartesian coordinates.

Q2:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 in Cartesian coordinates.

  • A 𝑥 + 𝑦 + 𝑧
  • B 1 8 ( 𝑥 + 𝑦 + 𝑧 )
  • C 6 ( 𝑥 + 𝑦 + 𝑧 )
  • D ( 𝑥 + 𝑦 + 𝑧 )
  • E 3 𝑥 + 𝑦 + 𝑧

Q3:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑒 in Cartesian coordinates.

  • A 𝑒 4 𝑥 + 4 𝑦 + 4 𝑧
  • B 𝑒 4 𝑥 4 𝑦 4 𝑧
  • C 2 𝑥 𝑒 2 𝑦 𝑒 2 𝑧 𝑒
  • D 𝑒 4 𝑥 + 4 𝑦 + 4 𝑧 6
  • E 𝑒 4 𝑥 + 4 𝑦 + 4 𝑧

Q4:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 in terms of the spherical coordinates 𝜌 , 𝜃 , and 𝜙 .

  • A 1 2 𝜌
  • B12
  • C 1 2 𝜌
  • D6
  • E 6 𝜌

Q5:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 .

  • A 5 𝑥
  • B 2 0 𝑥
  • C 5 𝑥
  • D 𝑥
  • E 2 0 𝑥

Q6:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑒 in terms of the spherical coordinates 𝜌 , 𝜃 , and 𝜙 .

  • A 2 1 1 𝜌 𝑒
  • B 4 𝜌 6 𝑒
  • C 4 1 1 𝜌 𝑒
  • D 4 𝜌 + 6 𝑒
  • E 4 1 + 1 𝜌 𝑒

Q7:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 + 𝑦 + 𝑧 .

  • A 𝑥 𝑥 + 𝑦 + 𝑧
  • B 1 2 𝑥 + 𝑦 + 𝑧
  • C 3 𝑥 𝑥 + 𝑦 + 𝑧
  • D 6 𝑥 𝑥 + 𝑦 + 𝑧
  • E 2 𝑥 𝑥 + 𝑦 + 𝑧

Q8:

Find the Laplacian of the function 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑒 in Cartesian coordinates.

  • A 1 3 𝑒
  • B 𝑒
  • C 6 𝑒
  • D 3 𝑒
  • E 1 6 𝑒

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