Surface Integrals Quiz

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Questions and Answers

What is the value of the surface integral $\oiint\limits_{S} \mathbf{n} \cdot \nabla\phi dS$, where $S$ is the surface of a sphere of unit radius and $\mathbf{n}$ is the outward unit normal vector on $S$, for the function $\phi = 3x^2 + 4y^2 + 5z^2$?

  • 24π
  • 72π (correct)
  • 12π
  • 48π

For the function $\phi = x^3 + y^3 + z^3$, what is the value of the surface integral $\oiint\limits_{S} \mathbf{n} \cdot \nabla\phi dS$, where $S$ is the surface of a sphere of unit radius and $\mathbf{n}$ is the outward unit normal vector on $S$?

  • 12π
  • 20π
  • 16π (correct)

If the function $\phi = 2x^2 + 3y^2 + 6z^2$, what is the value of the surface integral $\oiint\limits_{S} \mathbf{n} \cdot \nabla\phi dS$, where $S$ is the surface of a sphere of unit radius and $\mathbf{n}$ is the outward unit normal vector on $S$?

  • 36π
  • 24π (correct)
  • 48π
  • 12π

What is the Fourier series expansion of 𝑥 ଷ in the interval −1 ≤ 𝑥 < 1 with periodic continuation?

<p>An infinite sum of sines and cosines (D)</p> Signup and view all the answers

What is the period of the Fourier series expansion of 𝑥 ଷ in the interval −1 ≤ 𝑥 < 1?

<p>2 (D)</p> Signup and view all the answers

What is the range of the Fourier coefficients in the series expansion of 𝑥 ଷ?

<p>Complex numbers (D)</p> Signup and view all the answers

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