3D Delta Function
The 3D delta function, also known as the three-dimensional Dirac delta function, is a mathematical concept used in physics to represent a point source of a field in three-dimensional space. It is defined as zero everywhere except at the origin, where it is infinite in such a way that its integral over all space is equal to one.
3 Key excerpts on "3D Delta Function"
eBook - ePubFundamental Principles of Optical Lithography
The Science of Microfabrication
- Chris Mack(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Appendix C The Dirac Delta Function The Dirac delta function (also called the unit impulse function) is a mathematical abstraction which is often used to describe (i.e. approximate) some physical phenomenon. The main reason it is used has to do with some very convenient mathematical properties which will be described below. In optics, an idealized point source of light can be described using the delta function. Of course, real points of light will have finite width, but if the point is narrow enough, approximating it with a delta function can be very useful. C.1 Definition The Dirac delta function is in fact not a function at all, but a distribution (a generalized function, such as a probability distribution) that is also a measure (i.e. it assigns a value to a function)–terms that come from probability and set theory. However, for our purposes it will suffice to consider it a special function with infinite height, zero width and an area of 1. It can be considered the derivative of the Heaviside step function. To help think about the Dirac delta function, consider a rectangle with one side along the x-axis centered about x = x ο such that the area of the rectangle is 1 (this is equivalent to a uniform probability distribution). Obviously there are many such rectangles, as shown in Figure C.1. We can construct a Dirac delta function by starting with a square of height and width of 1. If we halve the width and double the height, the area will remain constant. We can repeat this process as many times as we wish. As the width goes to zero, the height will become infinite but the area will remain 1. Any unit area rectangle, centered at xo, can be expressed as Figure C.1 Geometrical construction of the Dirac delta function (C.1) where rect is the common rectangle function...
eBook - ePubMechanics of Flow-Induced Sound and Vibration, Volume 1
General Concepts and Elementary Sources
- William K. Blake(Author)
- 2017(Publication Date)
- Academic Press(Publisher)
...The component (∇× u) n is normal to the surface. This theorem in fluid dynamics relates vorticity to velocity. 1.6.4 Dirac Delta Function Dirac delta function, introduced previously in Eq. (1.36), is the most widely used generalized function in this book. It is defined formally as an integral ∫ − ∞ ∞ G (t) δ (t − t 0) d x = G (t 0) (1.62) (1.62) The integral’s limits may be finite and − T ≤ t 0 ≤ T. The delta function is also commonly regarded as a spike of indeterminate magnitude at t = t 0 but having an integral equal to unity. Thus we commonly see, e.g., Ref. [15 – 17] lim t → t 0 δ (t − t 0) = ∞ and δ (t) = 0 t ≠ 0 The above conditions imply the integral: ∫ 0 ∞ δ (t − t 0) d t = 1 (1.63) (1.63) The Fourier transform of the delta function. is F [ δ ] = 1 2 π ∫ − ∞ ∞ e i ω t δ (t − t 0) d t = 1 2 π e i ω t 0 (1.64) (1.64) The inverse transform then serves as an alternative definition of the delta function, which will be useful in future chapters, i.e. δ (t − t 0) = 1 2 π ∫ − ∞ ∞ e i ω (t 0 − t) d ω (1.65) (1.65) The multi-dimensional delta function has uses in formulating Green functions and depicting localized sources and force distributions. To this end, using Cartesian coordinates for example, δ (x − x 0) = δ (x − x 0) δ (y − y 0) δ (z − z 0) (1.66) (1.66) If x is contained in volume Δ V which also contains x 0, then ∫ ∫ ∫ Δ V δ (x − x 0) d 3 x = 1 (1.67) (1.67) and if x is not within Δ V,. then ∫ ∫ ∫ Δ V δ (x − x 0) d 3 x = 0 (1.68) (1.68) The Laplacian of 1 / (x − x 0) can be expressed as a Dirac delta function ∇ 2 (1 / (x − x 0)) = − 4 π δ (x − x 0) (1.69) (1.69) for which ∇ 2 (1 / (x − x 0)) = 0 for x ≠ x 0 and the volume integral. is ∫ ∫ ∫ Δ V ∇ 2 (1 / (x − x 0)) d 3 x = − 4 π (1.70) (1.70) References 1. Beranek L. Noise and vibration control New York: McGraw-Hill; 1971. 2. Bendat JS, Piersol AG. Random data analysis and measurement procedures 4th ed. New York: Wiley; 2010. 3...
eBook - ePub- Vadim Backman, Adam Wax, Hao F. Zhang(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...1 General Introductory Topics Part 1: Fundamental Mathematics Here, we introduce basic concepts in mathematics and statistics that will be useful in the coming chapters and experiments. The content is very limited and serves only to remind readers what should be known prior to continuing. For more detailed explanations and derivations, please refer to subject matter textbooks that focus on those topics. Dirac Delta Pulse There is often a need to consider the effect on a system by a forcing function that acts for a very short time period, such as a “kick” or an “impulse.” Such an impulse is called the Dirac delta (δ) pulse, shown in Figure 1.1, and is defined as δ (x) = { + ∞, x = 0 0, x ≠ 0 (1.1) Figure 1.1 Dirac delta pulse. Furthermore, the delta pulse satisfies an identity constraint: ∫ − ∞ + ∞ δ (x) d x = 1 (1.2) The Dirac delta is not a function, as any extended-real function that is equal to zero everywhere except one single point cannot have a total integral of 1. While it is more convenient to define the Dirac delta as a distribution, it may be manipulated as though it were a function, thus conferring its many properties and characteristics. The Dirac delta can be scaled by a nonzero scalar α : ∫ − ∞ + ∞ δ (α x) d x = ∫ − ∞ + ∞ δ (u) d u | α | = 1 | α | (1.3) so that δ (α x) = δ (x) | α | (1.4) The Dirac delta is symmetrical and follows an even distribution such that δ (− x) = δ (x) (1.5) The Dirac delta exhibits a translation property where the integral of a pulse delayed. by d returns the original function evaluated at d shown by ∫ − ∞ + ∞ f (x) δ (x − d) d x = f (d) (1.6) This is also called the sifting property of the delta pulse, as it “sifts out” the value f (d) from the function f (x)...
