Pre-order Ellie's new album 'Brightest Blue': Ellie Goulding - Flux LISTEN: More Ellie. Air Core with an Intensified Magnetic Field. The main purpose of the core is to contain the magnetic flux and create a well-defined, predictable path for the flux. This flux path, and the mean distance covered by the flux within the magnetic material, is defined as the Magnetic Path Length (MPL) (see Figure 3-2).
Flux (Surface Integrals of Vectors Fields)
Derivation of formula for Flux
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Suppose the velocity of a fluid in xyz space is described by the vector field F(x,y,z). Let S be a surface in xyz space. The flux across S is the volume of fluid crossing S perunit time. The figure below shows a surface S and the vector field F at variouspoints on the surface.
What is the formula for the flux? Let us consider the flux throughan infinitesimal piece of the surface with area dS above the point(x_0,y_0) in the xy plane. (The surface is denoted by the dotted region.)Let n denote the unit normal vector to the surface. Let ussuppose that the velocity vector F(x_0,y_0,z_0) of the fluid makesanangle theta with the unit normal. In one unit of time a blobof fluid of length |F|=|F(x_0,y_0,z_0)|will pass through the surface. (The blob is in the shape of a parallelepiped.)
The volume of this fluid is
The base area is dS. The height is equal to |F|cos(theta). Hence, we have
Here we have used the fact that |n|=1 and the property of thedotproduct. The total flux through the surface is
This is a surface integral.
We can write the above integral as an iterated double integral.Suppose that the surface S is described by the function z=g(x,y),where (x,y) lies in a region R of the xy plane. Theunit normalvector on the surface above (x_0,y_0) (pointing in the positivez direction) is
The surface area of an infinitesimalpiece of the surfaceabove a infinitesimal region in the xy plane with area dAcontaining (x_0,y_0) is given by
Hence, it follows that the total flux is
If we are asked for the flux in the negative z direction, thenwe use the vector <g_x(x,y),g_y(x,y),-1> for the normal direction.
Formula for Flux for Parametric Surfaces
Suppose that the surface S is described inparametric form:
where (u,v) lies in some region R of the uv plane. It can beshown that
Here, x means the cross product.Note, one may have to multiply the normal vector r_u x r_v by -1 to get the correctdirection.
Example Diskkeeper advanced cleaner 1 0 10.
Find the flux of the vector field <y,x,z> in the negative zdirection through the part of the surface z=g(x,y)=16-x^2-y^2 thatlies above the xy plane (see the figure below).
For this problem:
It follows that the normal vector is <-2x,-2y,-1>. ComputingFo<-2x,-2y,-1>, we have
Here we use the fact that z=16-x^2-y^2. Hence, the integralbecomes
The region R is the disk of radius 4 centered at the originshown below. It is convenient to convert topolar coordinates.The region of integration is 0<=r<=4 and 0<=theta<=2*pi.Substituting,
we have
Here we have used the fact that x^2+y^2=r^2. File juicer file extraction tool 4 79. (We can also integratewith respect to theta first, then r.)The inner integral with respect to r yields (remember theta is heldconstant)
which is equal to
The inner integral is
Hence, the flux through the surface in the downward z direction is-128*pi cubic units per unit time. This means a negative amount offluid crosses the surface in the downward direction. In other words,there is a flux of 128*pi cubic units per unit time in the upwardz direction.
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Flux is an innovative application for Mac OS X that will simplify the design and layout of web pages thanks to a visual editor for web pages that allows you to insert all kinds of elements just by dragging and dropping.
This can be used to generate static pages or dynamic pages that make use of technologies like PHP, AJAX or Ruby on Rails. The two languages that allows Flux to make a page or XHTML and CSS, so the resulting website can be accessed by practically any browser.
Flux also facilities the creation of styles and their subsequent application to any element on a page. The sidebar on the left lists all the styles in the style sheet relating to the project. Just drag the label on an item to apply the appropriate style you need.
Begin a project from scratch or part of the templates that Flux has available to familiarize yourself with the program.
This can be used to generate static pages or dynamic pages that make use of technologies like PHP, AJAX or Ruby on Rails. The two languages that allows Flux to make a page or XHTML and CSS, so the resulting website can be accessed by practically any browser.
Flux also facilities the creation of styles and their subsequent application to any element on a page. The sidebar on the left lists all the styles in the style sheet relating to the project. Just drag the label on an item to apply the appropriate style you need.
Begin a project from scratch or part of the templates that Flux has available to familiarize yourself with the program.
Flux 7 7 0 38
NotesFor Mac OS X 10.5 or later. Users of Mac OS X Tiger (10.4.9) must download Flux 1.4.91