A surface integral is an extension of the concept of a line integral to a two-dimensional surface. Surface integrals are used to calculate the accumulation of a scalar or vector quantity over a surface in three-dimensional space. They play a significant role in multivariable calculus and have applications in physics, engineering, and mathematics.
Surface integrals come in two main types: scalar surface integrals and vector surface integrals. Scalar surface integrals compute the accumulation of a scalar function over a surface, while vector surface integrals involve the dot product of a vector field and a differential area vector on the surface.
Applications of surface integrals include calculating the flux of a vector field through a surface, the surface area, and mass and center of mass of a lamina. Surface integrals are also connected to higher-dimensional analogs of the fundamental theorems of calculus, such as Stokes’ theorem and the divergence theorem.