Surface Integral Calculation Quiz

Surface Integrals

• The surface integral for surface area is denoted as $$\int_a^b\int_c^d |{\bf r}_u\times{\bf r}_v|,du,dv$$
• The integrand $|{\bf r}_u\times{\bf r}_v|,du,dv$ represents the area of a tiny parallelogram, which is a very small surface area
• This integrand can be abbreviated as $dS$, making the shortened version of the integral $$\dint{D} 1\cdot dS$$
• The $dS$ hides more detail than $dA$, which is used for integrating functions over regions in the plane

Surface Integrals in Space

• We can compute integrals over surfaces in space using $$\dint{D} f(x,y,z), dS$$
• This requires a vector function ${\bf r}(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle$ for the surface
• The integral we compute is $$\int_a^b\int_c^d f(x(u,v),y(u,v),z(u,v)) |{\bf r}_u\times{\bf r}_v|,du,dv$$

Partial Differential Equations

• A partial differential equation (PDE) is an equation that computes a function between various partial derivatives of a multivariable function
• The function is often thought of as an "unknown" to be solved for
• It is usually impossible to write down explicit formulae for solutions of partial differential equations
• Numerical approximations of solutions can be obtained using computers
• Pure mathematical research on partial differential equations focuses on identifying general qualitative features of solutions, such as existence, uniqueness, regularity, and stability