4: Mathematics Review

4: Mathematics Review B. Rouben McMaster University Course EP 4D03/6D03 Nuclear Reactor Analysis (Reactor Physics) 2014 Sept.-Dec.

Table of Contents Review of: • Co-ordinate systems (Cartesian, Cylindrical, Spherical) • Differential lengths, surface areas, volumes • Integrating over solid angle • Gradient, Divergence, Laplacian • Leakage out of a volume and Gauss’ theorem

Co-Ordinate Systems A point in 3-dimensional space can be identified in a number of different co-ordinate systems. The ones which will be most useful to us are: • Cartesian: here the co-ordinates are the familiar x, y, z • Cylindrical: here the co-ordinates are r,  , z (see diagram on next slide) • Spherical: here the co-ordinates are r,  ,  (or )(see diagram on 2nd next slide). Sometimes it is convenient to use =cos as variable instead of .

Cylindrical Co-Ordinates z is the height of the point above the x-y plane; r is the distance from the origin to the point’s projection on the x-y plane;  (I sometimes use ) is the angle of rotation from the x-axis to the point’s projection on the x-y plane

Polar Co-Ordinate System for Solid Angle

Spherical Co-Ordinates r is the distance of the point from the origin;  is the polar angle (“latitude” of point from z-axis, 0 to 180 deg);  (I use ) is the angle of rotation from the x-axis to the point’s projection on the x-y plane (similar to  in cylindrical co-ordinates)

Differential Lengths (Distances) Differential lengths ds corresponding to differentials in the various co-ordinates are needed for use in 1-d integrals in the various systems. They can be easily evaluated as the distances corresponding to changes in the various directions. • In Cartesian co-ordinates, ds = dx, dy, dz • In cylindrical co-ordinates, ds = dr, rd, dz (Note: I sometimes use  instead of ) • In spherical co-ordinates, ds= dr, rd, rsind (Note: I sometimes use  instead of ).

Differential Surface Areas • Differential surface areas perpendicular to the various directions (see some in next slide) are: • In Cartesian co-ordinates, perpendicular to directions along changes in x, y, z: dydz, dzdx, dxdy • In cylindrical co-ordinates, perpendicular to directions along changes in r, , z: rddz, drdz, rdrd • In spherical co-ordinates, perpendicular to directions along changes in r, ,  (or ): r2sindd, rsindrd, rdrd

Differential Surface Areas

Differential Volumes Differential volumes dV are needed in 3-d integrals in the various co-ordinate systems. They are (see next slide for drawings): • In Cartesian co-ordinates, dV = dxdydz • In cylindrical co-ordinates, dV= rdrddz (Note: I sometimes use  instead of ) • In spherical co-ordinates, dV= rsindrddr = r2sin d drd = (-)r2drdd (Note: I sometimes use  instead of )

Differential Volumes

Spherical Co-Ordinates:  Instead of  • Note: In spherical co-ordinates, it is often very useful, especially when evaluating integrals, to use   cos  as the variable, instead of the (“latitude”) variable . • This is so because d = - sind, so that the more complicated quantity on the right can easily be replaced by the simpler differential d. • The range  = 0 to  is replaced by the range  = -1 to +1 (when expressed in this order instead of +1 to -1, it removes the minus sign).

Integrating Over Solid Angle • Integrating over a solid angle is essentially equivalent to integrating over the surface of a unit sphere. • We must be aware of the range of solid angle (and therefore, of  and ) that we need to consider.

Integrating Over All Solid Angles

Neutrons Crossing Unit Area of a Plane For neutrons crossing the plane from below,  is between 0 and /2 ( = 0 to 1). For neutrons crossing from above to below,  is between /2 and  ( = -1 to 0). In this case the “number” will be negative.

Derivative of a Function • The derivative of a function f of a single variable (x) is the rate of change of f with respect to x, • The change fin the value of f when x changes by a small amountx can then be approximated by:

Gradient • If we now consider a function f of 3 variables (x, y, z), then the value of f will (in general) change if there is a change in any of the 3 variables. • We can define the directional rates of change with respect to each variable separately: • And the change in the value of f when there are small changes in all the variables can be approximated by

Gradient (cont’d) • The gradient of f can be thought of as a vector.

Gradient (cont’d) • If we write the increments in the independent variables of f also as a vector in the general direction , then the increment in f can be written as a dot product of the two vectors:

Physical Meaning of Gradient • If we imagine the increment in the independent variables to be a unit change in the direction then we can see that the increment in f, is simply the projection of f in the direction (see next slide). This projection will be largest if the direction of is the same as that of the vector . • This tells us that is the rate of change of f in the direction in which it increases most rapidly!

The Increment in f as a Projection B A

The Gradient Operator • In the previous slides, we defined the gradient of a function f. • We can think of that gradient as the action of a gradient operator on the function f. In this interpretation, the gradient operator in Cartesian co-ordinates is written as • We can see that the gradient operator is a vector operator.

Gradient Operator in Other Co-Ordinates From our knowledge of the differential lengths corresponding to the various variables (see prior slide), we can write the components of the gradient operator in other co-ordinates:

The Divergence of a Vector • In the previous slides, we saw that the gradient operator could be defined as a vector operator. • We saw the action of the gradient operator on a scalar function f. • But we can also define the action of the gradient operator can also operate on a vector function, . • This action is defined as the divergence of the function, and is a dot product, written as . • In Cartesian co-ordinates, the divergence is:

Physical Meaning of Divergence • The physical meaning of the divergence is that it is the “leakage” of the vector function out of a an infinitesimal volume around the point where the divergence is calculated, divided by the infinitesimal volume. • See proof of this “divergence theorem” in next slide. • Note: The proof is given in Cartesian co-ordinates, but holds for any shape of the infinitesimal volume.

Divergence Theorem for Vector (Current) x x+dx Infinitesimal volume; sides perpendicular to paper are dy and dz

Divergence in Other Co-Ordinates If we write the vector function F in terms of its components in other co-ordinates, the divergence operator becomes:

Leakage out of a Finite Volume • Subdivide a finite volume into infinitesimal subvolumes; apply the divergence theorem in each subvolume, and “add” all (i.e., integrate). • The “internal” leakages (across internal surfaces out of one subvolume and into a neighbouring subvolume) obviously cancel out, leaving only the leakage out of the external surface. • Therefore the net leakage out of the finite volume = the volume integral of the divergence of the current.

Leakage out of a Finite Volume • We have just proved Gauss’s famous Theorem:

Laplacian • The Laplacian of a function f , denoted , is defined as the divergence of the gradient of f : • The Laplacian of the flux  is useful in reactor physics, because of the Divergence Theorem and an approximation (Fick’s Law), , which says that the net current is proportional to the negative of the gradient of the flux , i.e., that the net neutron current flows from areas of high flux to areas of low flux.

Laplacian • The Fick’s Law approximation allows us to simplify to -D2 , where D is the constant of proportionality between the flux gradient and the current. • We consequently need the formulation of the Laplacian of the flux in the various co-ordinate systems. These are given in their most general form in the next slide.

Laplacian of the Flux

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4: Mathematics Review