![]() It can also be written in terms of a surface integral of a scalar field, where n^ is the normal vector to the surface. We have a vector field u → defined on the surface, the surface integral of the vector field is defined as: ∫ S u → ∙ dS → If the scalar field f(r) = 1, then the surface integral just gives us the area of the surface. We compute this integral by parameterizing the surface, by actually parameterizing r. The surface integral of the scalar field will be obtained by a Riemann sum, where we break the surface into small surface elements dS. Then multiply each element dS by the average value of the scalar field f, and finally we sum over all of those surface elements. We have some surface in three-dimensions. The surface itself is a two-dimensional object. We have a scalar field f defined on the surface. The work is actually line integral of the force along the curve, it is done by the force on the mass results in the change in the kinetic energy of the mass. Work in a physicist definition is the energy transferred to an object by a force. Usually we do the integral by parameterizing the curve. It’s basically saying that you have to take the component of u → along the curve to do the line integral: ∫ c u → ∙ dr →īut this actually can be written in terms of a line integral of a scalar field, where t^ is the tangent unit vector: ∫ c (u → ∙ t^) ds At the same time we have an element of the curve dr →, pointing and along the direction of the curve. On this curve, we have a well-defined vector field u →. Most of the physical problems that we define in terms of line integrals are actually line integrals of a vector field. Curve is parameterized x = x(t), y = y(t).There are 2 method of solving this integral: Then the integral is Riemann sum: ∫ c f(r) ds To do a line integral, we break the curve into small pieces ds, you have a small element of length ds and a value of f on that element, we multiply them together, and then we sum over all of the elements on the curve C. In Figure 12.9.5 you can select between five different vector fields.We have a curve C in the x-y plane, we can represent a point on this curve then by a vector r. In this activity, you will compare the net flow of different vector fields through our sample surface. Thus, the net flow of the vector field through this surface is positive. If we define a positive flow through our surface as being consistent with the yellow vector in Figure 12.9.4, then there is more positive flow (in terms of both magnitude and area) than negative flow through the surface. If we have a parametrization of the surface, then the vector \(\vr_s \times \vr_t\) varies smoothly across our surface and gives a consistent way to describe which direction we choose as “through” the surface. 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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