Surface integral

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics.

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics. = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere, where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The surface integral can also be expressed in the equivalent form \iint_S f \,\mathrm dS = \iint_T \left\|{\partial \mathbf{r} \over \partial x} \times {\partial \mathbf{r} \over \partial y}\right\| \mathrm dx\, \mathrm dy where . So that {\partial \mathbf{r} \over \partial x}=(1, 0, f_x(x,y)) , and {\partial \mathbf{r} \over \partial y}=(0, 1, f_y(x,y)) . So, \begin{align} A &{} = \iint_T \left\|\left(1, 0, {\partial f \over \partial x}\right)\times \left(0, 1, {\partial f \over \partial y}\right)\right\| \mathrm dx\, \mathrm dy \\ &{} = \iint_T \left\|\left(-{\partial f \over \partial x}, -{\partial f \over \partial y}, 1\right)\right\| \mathrm dx\, \mathrm dy \\ &{} = \iint_T \sqrt{\left({\partial f \over \partial x}\right)^2+\left({\partial f \over \partial y}\right)^2+1}\, \, \mathrm dx\, \mathrm dy \end{align} which is the standard formula for the area of a surface described this way. One can recognize the vector in the second-last line above as the normal vector to the surface. Because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space. This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface. ==Surface integrals of vector fields== Consider a vector field v on a surface S, that is, for each in S, v(r) is a vector. The integral of v on S was defined in the previous section. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface. For example, imagine that we have a fluid flowing through S, such that v(r) determines the velocity of the fluid at r. The flux is defined as the quantity of fluid flowing through S per unit time. This illustration implies that if the vector field is tangent to S at each point, then the flux is zero because, on the surface S, the fluid just flows along S, and neither in nor out. This also implies that if v does not just flow along S, that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s , which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. We find the formula \begin{align} \iint_S {\mathbf v}\cdot\mathrm d{\mathbf {s}} &= \iint_S \left({\mathbf v}\cdot {\mathbf n}\right)\,\mathrm ds\\ &{}= \iint_T \left({\mathbf v}(\mathbf{r}(s, t)) \cdot {\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t} \over \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\|}\right) \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\| \mathrm ds\, \mathrm dt\\ &{}=\iint_T {\mathbf v}(\mathbf{r}(s, t))\cdot \left(\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right) \mathrm ds\, \mathrm dt. \end{align} The cross product on the right-hand side of the last expression is a (not necessarily unital) surface normal determined by the parametrisation. This formula defines the integral on the left (note the dot and the vector notation for the surface element). We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface. This is equivalent to integrating \left\langle \mathbf{v}, \mathbf{n} \right\rangle \mathrm dS over the immersed surface, where \mathrm dS is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface. \omega=\omega_1 \mathrm dy \wedge \mathrm dz + \omega_2 \mathrm dz \wedge \mathrm dx + \omega_3 \mathrm dx \wedge \mathrm dy be a differential 2-form defined on a surface S in \mathbb{R}^3 , and let \mathbf{r}: D \to \mathbb{R}^3 be an orientation preserving parametrization of S with (s,t) in D. The coordinate 1-forms pull back as \mathbf{r}^*(\mathrm dx)=\mathrm d(x \circ \mathbf{r})=\frac{\partial(x \circ \mathbf{r})}{\partial s}\mathrm ds+\frac{\partial(x \circ \mathbf{r})}{\partial t}\mathrm dt So \mathbf{r}^*(\mathrm dx \wedge \mathrm dy) = \frac{\partial(x \circ \mathbf{r},y \circ \mathbf{r})}{\partial(s,t)} \mathrm ds \wedge \mathrm dt , where the notation on the right denotes the Jacobian determinant. The pullbacks of the other forms are similar. Then, the surface integral of \omega on S is given by \int_S \omega = \int_D \mathbf{r}^*(\omega) =\int_D \left[ (\omega_1\circ\mathbf{r})\frac{\partial(y\circ\mathbf r,z\circ\mathbf r)}{\partial(s,t)} + (\omega_2\circ\mathbf{r})\frac{\partial(z\circ\mathbf r,x\circ\mathbf r)}{\partial(s,t)} + (\omega_3\circ\mathbf{r})\frac{\partial(x\circ\mathbf r,y\circ\mathbf r)}{\partial(s,t)}\right] \mathrm ds \wedge \mathrm dt where {\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y\circ\mathbf r,z\circ\mathbf r)}{\partial(s,t)}, \frac{\partial(z\circ\mathbf r,x\circ\mathbf r)}{\partial(s,t)}, \frac{\partial(x\circ\mathbf r,y\circ\mathbf r)}{\partial(s,t)}\right) is the surface element normal to S. Let us note that the integral of this 2-form is the same as the surface integral of the vector field which has as components \omega_1 , \omega_2 and \omega_3 . ==References==

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