# Surface Integral **The [[integral]] of a multivariable function over a surface as the domain of integration.** ![[SurfaceIntegral.svg|600]] *A surface of integration divided into differential patches $dS$ with sides lengths $du$ and $dv$; $u$ and $v$ are variables from the parameterisation of the surface* A *surface integral* is the integral of a multivariable function, whether a scalar or a vector field, as evaluated *over a surface*. It is a generalisation of [[Multiple Integral|multiple integrals]] over arbitrary surfaces. Surface integrals use the [[cross product]] of the [[Partial Derivative|partial derivatives]] of a differential patch of the surface. This is known as the *surface element*. Over a scalar field, a surface integral is the *sum of all field values* on the surface multiplied by the magnitude surface element. For a vector field, it is instead the sum of the [[dot product]] of the *normal components* of all vectors passing through the field with the surface element. It is common to *parameterise* the surface in terms of two variables, defining a new curvilinear coordinate system representing the surface. This allows of the evaluation of the integral as though it is a double integral. A surface integral over a flat surface is the same as a double integral, although with different notation. It is the double integral analogue of a [[line integral]]. ## Definition ### Scalar field > [!NOTE] Surface integral of a scalar field > For some scalar field $f$, the *surface integral* on a surface $S$ is defined as > $\iint_{S}f\;dS=\iint_{T}f(\mathbf{r}(u,v))\left\|{\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}}\right\|\;du\;dv$ > where $\mathbf{r}(u, v)$ is a *parameterisation* of the surface onto the *parameter domain* $T$. The scalar field $f$ is the *integrand*, the surface $S$ is the *domain of integration*, and $dS$ represents a *differential patch* on the surface. The value of the surface integral of a scalar field *does not depend* on the parameterisation used. ### Vector field > [!NOTE] Surface integral of a vector field > For some vector field $\mathbf{F}$, the *surface integral* on a surface $S$ is defined as > $\iint_{S}\mathbf{F}\cdot d\mathbf{s}=\iint_{T}\mathbf{F}(\mathbf{r}(u,v))\cdot\left(\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right)\;du\;dv$ > where $\mathbf{r}(u, v)$ is a *parameterisation* of the surface onto the *parameter domain* $T$. The absolute value of the surface integral of a vector field is *dependent* on the parameterisation used. Intuitively, the surface integral of a vector field is the sum of the normal components of all vectors passing through the curve multiplied by the area of each differential patch.