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Higher Mathematics

Integral Calculus of Several Variables

This chapter systematically studies multiple integrals, including double integrals, triple integrals, line integrals, surface integrals, and their applications in geometry and physics.

Double and Triple Integrals

Definition and Properties of Double Integrals

  • ∬Df(x,y)dA\iint_D f(x, y) dA
  • Computation in rectangular and polar coordinates
  • Linearity, additivity over regions

Definition and Properties of Triple Integrals

  • ∭Ωf(x,y,z)dV\iiint_\Omega f(x, y, z) dV
  • Computation in rectangular, cylindrical, and spherical coordinates

Applications of Double and Triple Integrals

  • Area, volume, mass, and physical quantities

Line Integrals and Surface Integrals

First Kind Line Integral

  • ∫Cf(x,y)ds\int_C f(x, y) ds
  • Used to compute curve length, mass, etc.

Second Kind Line Integral

  • ∫CPdx+Qdy\int_C P dx + Q dy
  • Physical applications: work, flow

Surface Integrals

  • ∬Sf(x,y,z)dS\iint_S f(x, y, z) dS
  • ∬SPdydz+Qdzdx+Rdxdy\iint_S P dy dz + Q dz dx + R dx dy

Important Formulas

Green’s Theorem

  • ∬D(∂Q∂x−∂P∂y)dA=∫∂DPdx+Qdy\iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA = \int_{\partial D} P dx + Q dy

Gauss’s Theorem (Divergence Theorem)

  • ∭Ω∇⋅F⃗dV=∬∂ΩF⃗⋅dS⃗\iiint_\Omega \nabla \cdot \vec{F} dV = \iint_{\partial \Omega} \vec{F} \cdot d\vec{S}

Stokes’ Theorem

  • ∬S(∇×F⃗)⋅dS⃗=∫∂SF⃗⋅dr⃗\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \int_{\partial S} \vec{F} \cdot d\vec{r}

Exercises

  1. Compute the double integral ∬Dx2ydA\iint_D x^2 y dA, where DD is 0≤x≤1,0≤y≤20 \leq x \leq 1, 0 \leq y \leq 2.
  2. Compute the triple integral ∭ΩzdV\iiint_\Omega z dV, where Ω\Omega is 0≤x≤1,0≤y≤1,0≤z≤20 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 2.
  3. Compute the line integral ∫C(x2+y2)ds\int_C (x^2 + y^2) ds, where CC is the unit circle x2+y2=1x^2 + y^2 = 1.
  4. Use Green’s theorem to compute ∫C(x−y)dx+(x+y)dy\int_C (x - y) dx + (x + y) dy, where CC is the positively oriented circle centered at the origin with radius 1.
  5. Compute the surface integral ∬SzdS\iint_S z dS over the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1.
Reference Answers

1. Compute the double integral ∬Dx2ydA\iint_D x^2 y dA

∫01∫02x2ydydx=∫01x2[12y2]02dx=∫01x2×2dx=2∫01x2dx=2×13=23\int_0^1 \int_0^2 x^2 y dy dx = \int_0^1 x^2 [\frac{1}{2}y^2]_0^2 dx = \int_0^1 x^2 \times 2 dx = 2 \int_0^1 x^2 dx = 2 \times \frac{1}{3} = \frac{2}{3}

2. Compute the triple integral ∭ΩzdV\iiint_\Omega z dV

∫01∫01∫02zdzdydx=∫01∫01[12z2]02dydx=∫01∫012dydx=2\int_0^1 \int_0^1 \int_0^2 z dz dy dx = \int_0^1 \int_0^1 [\frac{1}{2}z^2]_0^2 dy dx = \int_0^1 \int_0^1 2 dy dx = 2

3. Compute the line integral ∫C(x2+y2)ds\int_C (x^2 + y^2) ds

x2+y2=1x^2 + y^2 = 1, dsds along the unit circle, the value is 2π2\pi

4. Use Green’s theorem to compute ∫C(x−y)dx+(x+y)dy\int_C (x - y) dx + (x + y) dy

∂(x+y)∂x−∂(x−y)∂y=1−(−1)=2\frac{\partial (x + y)}{\partial x} - \frac{\partial (x - y)}{\partial y} = 1 - (-1) = 2

The area is π\pi, so the value is 2π2\pi

5. Compute the surface integral ∬SzdS\iint_S z dS over the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1

The sphere is symmetric, zz is an odd function about z=0z=0, so the integral is 00