6 Questions
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 $\phi = 3x^2 + 4y^2 + 5z^2$?
72π
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 $S$?
16π
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 $S$?
24π
What is the Fourier series expansion of 𝑥 ଷ in the interval −1 ≤ 𝑥 < 1 with periodic continuation?
An infinite sum of sines and cosines
What is the period of the Fourier series expansion of 𝑥 ଷ in the interval −1 ≤ 𝑥 < 1?
2
What is the range of the Fourier coefficients in the series expansion of 𝑥 ଷ?
Complex numbers
Test your knowledge of surface integrals in three-dimensional Cartesian space with this quiz. Calculate the surface integral of a given function over the surface of a unit sphere using the outward unit normal vector. Sharpen your skills in vector calculus and surface area calculations.
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