# Kinematics of surface integrals question

• skybobster
In summary, this conversation discusses a question in the field of continuum mechanics and fluid dynamics regarding the time rate of change of a surface integral of a vector field. This integral is known by different names in various applications, such as "Faraday's law for moving media" or "Zorawski's criterion." The conversation also mentions the use of two vector fields, one for the velocity and the other for a specified field, like the magnetic field in electromagnetism. The derivations for this integral are typically based on a surface patch surrounded by a closed curve, with the integral vanishing in certain cases. The speaker's question is whether this also applies to closed moving surfaces, such as a deformed sphere, and why it should be
skybobster
This is a continuum mechanics/fluid dynamics question concerning the time rate of change of a surface integral of a vector field, where the surface is flowing along in a velocity field (like in a fluid). (Gauss's law is for fixed surfaces.) This integral goes by various names in different applications, like "Faraday's law for moving media" in electromagnetism, or "Zorawski's criterion" in fluid dynamics (when the integral vanishes). (Truesdell, "Kinematics of Vorticity" page 55). We have two vector fields, one is the velocity vector field, and the other is some other specified vector field, like the magnetic field in E-M. Most of the derivations are based on a surface patch flowing along in the fluid (or more generally a velocity field), this patch surrounded by a closed curve. When the integral vanishes, it means that the integral of the given vector field dotted into the moving surface element of the (velocity) vector-tube (the tube swept out by the flowing surface patch) for the given vector field remains the same during the motion, and we end up with an elegant vector equation (called sometimes Zorawski's criterion). My question is this: Is this also true for closed moving surfaces (instead of a surface patch) like for example a deformed sphere flowing along? Nowhere in the literature can I find this important case of a closed moving surface.

## 1. What is the definition of a surface integral?

A surface integral is a mathematical concept used to calculate the flux or flow of a vector field across a surface. It involves breaking the surface into smaller elements and integrating the vector function over each element to find the total flux.

## 2. How is a surface integral different from a regular integral?

A regular integral involves finding the area under a curve in two dimensions, while a surface integral involves finding the flux through a surface in three dimensions. Additionally, a regular integral has a single variable, while a surface integral has two variables (u and v) to describe the surface.

## 3. What is the significance of the orientation of a surface in a surface integral?

The orientation of the surface determines the direction of the flux. If the surface is oriented in the same direction as the vector field, the flux will be positive. If the surface is oriented in the opposite direction, the flux will be negative. It is important to specify the orientation when setting up a surface integral.

## 4. How is a surface integral calculated?

A surface integral can be calculated using a double integral, where the limits of integration are determined by the parametric equations of the surface. The integrand is the dot product of the vector field and the surface normal vector at each point on the surface.

## 5. In what real-life situations is the concept of surface integrals used?

Surface integrals have many applications in physics and engineering, such as calculating the flow of fluids across a surface, determining the electric field around a charged object, or finding the mass of an object with a varying density. They are also used in computer graphics to render three-dimensional objects and in geographical mapping to calculate surface areas and volumes.

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