In other words, we will need to pay attention to the direction in which these vectors move through our surface and not just the magnitude of the green vectors.
Notice that some of the green vectors are moving through the surface in a direction opposite of others. In order to measure the amount of the vector field that moves through the plotted section of the surface, we must find the accumulation of the lengths of the green vectors in Figure 12.9.4. The decomposition of three-dimensional vector field evaluated along a surface into normal and tangent components The component that is tangent to the surface is plotted in purple. One component, plotted in green, is orthogonal to the surface. In the next figure, we have split the vector field along our surface into two components. Any portion of our vector field that flows along (or tangent) to the surface will not contribute to the amount that goes through the surface. The central question we would like to consider is “How can we measure the amount of a three dimensional vector field that flows through a particular section of a curved surface?”, so we only need to consider the amount of the vector field that flows through the surface. A three-dimensional vector field evaluated along a surface So instead, we will look at Figure 12.9.3.
We don't care about the vector field away from the surface, so we really would like to just examine what the output vectors for the \((x,y,z)\) points on our surface. A three-dimensional vector field and a surface We are interested in measuring the flow of the fluid through the shaded surface portion. There is also a vector field, perhaps representing some fluid that is flowing. We have a piece of a surface, shown by using shading. In Figure 12.9.2, we illustrate the situation that we wish to study in the remainder of this section. \newcommand\) Subsection 12.9.1 The Idea of the Flux of a Vector Field through a Surface
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