Surface integrals are like line integrals, but for 3D surfaces. They measure how a function adds up over a surface, useful for finding mass, heat flow, or fluid movement across curved areas.

To calculate surface integrals, we use special math tricks. We can break the surface into smaller pieces or project it onto a flat plane. This helps us solve problems in physics and engineering involving curved surfaces.

Understanding Surface Integrals

Definition of surface integrals

• Surface integrals extend line integrals to surfaces in 3D space measuring accumulation of scalar function over surface
• Represented as $\iint_S f(x,y,z) dS$ analogous to finding mass of thin shell with varying density
• Can represent total flux, heat flow, or other physical quantities across surface (heat dissipation, fluid flow)

Evaluation of surface integrals

• Parameterize surfaces as $\mathbf{r}(u,v) = (x(u,v), y(u,v), z(u,v))$ using parameter bounds to define integration region
• Evaluate by:
  1. Rewrite integrand in terms of u and v
  2. Compute surface element $dS = |\mathbf{r}_u \times \mathbf{r}_v| du dv$
  3. Set up iterated integral: $\iint_S f(x,y,z) dS = \int_a^b \int_c^d f(\mathbf{r}(u,v)) |\mathbf{r}_u \times \mathbf{r}_v| du dv$
• Alternative method projects surface onto coordinate plane using $dS = \sqrt{1 + (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2} dA$ for z = f(x,y)

Applications of surface integrals

• Calculate surface area by setting $f(x,y,z) = 1$ and evaluating $\iint_S dS$
• Find average value of function over surface using $\frac{1}{A} \iint_S f(x,y,z) dS$, where A is surface area
• Determine center of mass of surface with varying density or total heat flow through surface with varying temperature

Surface integrals for vector fields

• Flux integrals measure flow of vector field through surface defined as $\iint_S \mathbf{F} \cdot \mathbf{n} dS$, where $\mathbf{n}$ is unit normal vector
• Compute using surface parameterization: $\iint_S \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) du dv$
• For surfaces z = f(x,y): $\iint_S \mathbf{F} \cdot \langle -f_x, -f_y, 1 \rangle dA$
• Applications include fluid flow through surface and electric or magnetic flux (Faraday's law, Gauss's law)
• Connects to divergence theorem relating surface integral to volume integral of divergence