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# Module direction_cosine

source code

 Functions

 calc_spheroid_di(data, diff_data) Function for calculating the direction cosine dz. source code

 calc_spheroid_ddi(data, diff_data) Function for calculating the partial derivatives of the direction cosine dz. source code

 calc_spheroid_d2di(data, diff_data) Function for calculating the second partial derivatives of the direction cosine dz. source code

 calc_ellipsoid_di(data, diff_data) Function for calculating the direction cosines dx, dy, and dz. source code

 calc_ellipsoid_ddi(data, diff_data) Function for calculating the partial derivatives of the direction cosines dx, dy, and dz. source code

 calc_ellipsoid_d2di(data, diff_data) Function for calculating the second partial derivatives of the direction cosines dx, dy, dz. source code
 Variables
__package__ = `'lib.diffusion'`

Imports: cos, sin, dot

 Function Details

### calc_spheroid_di(data, diff_data)

source code

Function for calculating the direction cosine dz.

dz is the dot product between the unit bond vector and the unit vector along Dpar and is given by:

```   dz = XH . Dpar.
```

The unit Dpar vector is:

```            | sin(theta) * cos(phi) |
Dpar  =  | sin(theta) * sin(phi) |
|      cos(theta)       |
```

### calc_spheroid_ddi(data, diff_data)

source code

Function for calculating the partial derivatives of the direction cosine dz.

The theta partial derivative of the unit Dpar vector is:

```   dDpar      | cos(theta) * cos(phi) |
------  =  | cos(theta) * sin(phi) |
dtheta     |      -sin(theta)      |
```

The phi partial derivative of the unit Dpar vector is:

```   dDpar     | -sin(theta) * sin(phi) |
-----  =  |  sin(theta) * cos(phi) |
dphi      |           0            |
```

O is the orientational parameter set {theta, phi}

### calc_spheroid_d2di(data, diff_data)

source code

Function for calculating the second partial derivatives of the direction cosine dz.

The theta-theta second partial derivative of the unit Dpar vector is:

```   d2Dpar      | -sin(theta) * cos(phi) |
-------  =  | -sin(theta) * sin(phi) |
dtheta2     |      -cos(theta)       |
```

The theta-phi second partial derivative of the unit Dpar vector is:

```     d2Dpar        | -cos(theta) * sin(phi) |
-----------  =  |  cos(theta) * cos(phi) |
dtheta.dphi     |           0            |
```

The phi-phi second partial derivative of the unit Dpar vector is:

```   dDpar     | -sin(theta) * cos(phi) |
-----  =  | -sin(theta) * sin(phi) |
dphi2     |           0            |
```

O is the orientational parameter set {theta, phi}

### calc_ellipsoid_di(data, diff_data)

source code

Function for calculating the direction cosines dx, dy, and dz.

# Direction cosines

dx is the dot product between the unit bond vector and the unit vector along Dx. The equation is:

```   dx = XH . Dx
```

dy is the dot product between the unit bond vector and the unit vector along Dy. The equation is:

```   dy = XH . Dy
```

dz is the dot product between the unit bond vector and the unit vector along Dz. The equation is:

```   dz = XH . Dz
```

# Unit vectors

The unit Dx vector is:

```          | -sin(alpha) * sin(gamma) + cos(alpha) * cos(beta) * cos(gamma) |
Dx  =  | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
|                    cos(alpha) * sin(beta)                      |
```

The unit Dy vector is:

```          | cos(alpha) * sin(gamma) + sin(alpha) * cos(beta) * cos(gamma) |
Dy  =  | cos(alpha) * cos(gamma) - sin(alpha) * cos(beta) * sin(gamma) |
|                   sin(alpha) * sin(beta)                      |
```

The unit Dz vector is:

```          | -sin(beta) * cos(gamma) |
Dz  =  |  sin(beta) * sin(gamma) |
|        cos(beta)        |
```

### calc_ellipsoid_ddi(data, diff_data)

source code

Function for calculating the partial derivatives of the direction cosines dx, dy, and dz.

The alpha partial derivative of the unit Dx vector is:

```    dDx       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
dalpha     |                   -sin(alpha) * sin(beta)                      |
```

The beta partial derivative of the unit Dx vector is:

```    dDx      | -cos(alpha) * sin(beta) * cos(gamma) |
-----  =  |  cos(alpha) * sin(beta) * sin(gamma) |
dbeta     |       cos(alpha) * cos(beta)         |
```

The gamma partial derivative of the unit Dx vector is:

```    dDx       | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
------  =  |  sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
dgamma     |                             0                                  |
```

The alpha partial derivative of the unit Dy vector is:

```    dDy       | -sin(alpha) * sin(gamma) + cos(alpha) * cos(beta) * cos(gamma) |
------  =  | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
dalpha     |                    cos(alpha) * sin(beta)                      |
```

The beta partial derivative of the unit Dy vector is:

```    dDy      | -sin(alpha) * sin(beta) * cos(gamma) |
-----  =  |  sin(alpha) * sin(beta) * sin(gamma) |
dbeta     |       sin(alpha) * cos(beta)         |
```

The gamma partial derivative of the unit Dy vector is:

```    dDy       |  cos(alpha) * cos(gamma) - sin(alpha) * cos(beta) * sin(gamma) |
------  =  | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
dgamma     |                             0                                  |
```

The alpha partial derivative of the unit Dz vector is:

```    dDz       | 0 |
------  =  | 0 |
dalpha     | 0 |
```

The beta partial derivative of the unit Dz vector is:

```    dDz      | -cos(beta) * cos(gamma) |
-----  =  |  cos(beta) * sin(gamma) |
dbeta     |        -sin(beta)       |
```

The gamma partial derivative of the unit Dz vector is:

```    dDz       | sin(beta) * sin(gamma) |
------  =  | sin(beta) * cos(gamma) |
dgamma     |           0            |
```

### calc_ellipsoid_d2di(data, diff_data)

source code

Function for calculating the second partial derivatives of the direction cosines dx, dy, dz.

# Dx Hessian

The alpha-alpha second partial derivative of the unit Dx vector is:

```    d2Dx       | sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
-------  =  | sin(alpha) * cos(gamma) + cos(alpha) * cos(beta) * sin(gamma) |
dalpha2     |                  -cos(alpha) * sin(beta)                      |
```

The alpha-beta second partial derivative of the unit Dx vector is:

```       d2Dx         |  sin(alpha) * sin(beta) * cos(gamma) |
------------  =  | -sin(alpha) * sin(beta) * sin(gamma) |
dalpha.dbeta     |      -sin(alpha) * cos(beta)         |
```

The alpha-gamma second partial derivative of the unit Dx vector is:

```       d2Dx          | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
-------------  =  |  cos(alpha) * sin(gamma) + sin(alpha) * cos(beta) * cos(gamma) |
dalpha.dgamma     |                             0                                  |
```

The beta-beta second partial derivative of the unit Dx vector is:

```    d2Dx      | -cos(alpha) * cos(beta) * cos(gamma) |
------  =  |  cos(alpha) * cos(beta) * sin(gamma) |
dbeta2     |      -cos(alpha) * sin(beta)         |
```

The beta-gamma second partial derivative of the unit Dx vector is:

```       d2Dx         | cos(alpha) * sin(beta) * sin(gamma) |
------------  =  | cos(alpha) * sin(beta) * cos(gamma) |
dbeta.dgamma     |                 0                   |
```

The gamma-gamma second partial derivative of the unit Dx vector is:

```    d2Dx       | sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
-------  =  | sin(alpha) * cos(gamma) + cos(alpha) * cos(beta) * sin(gamma) |
dgamma2     |                            0                                  |
```

# Dy Hessian

The alpha-alpha second partial derivative of the unit Dy vector is:

```    d2Dy       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
-------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
dalpha2     |                   -sin(alpha) * sin(beta)                      |
```

The alpha-beta second partial derivative of the unit Dy vector is:

```       d2Dy         | -cos(alpha) * sin(beta) * cos(gamma) |
------------  =  |  cos(alpha) * sin(beta) * sin(gamma) |
dalpha.dbeta     |       cos(alpha) * cos(beta)         |
```

The alpha-gamma second partial derivative of the unit Dy vector is:

```       d2Dy          | -sin(alpha) * cos(gamma) - cos(alpha) * cos(beta) * sin(gamma) |
-------------  =  |  sin(alpha) * sin(gamma) - cos(alpha) * cos(beta) * cos(gamma) |
dalpha.dgamma     |                             0                                  |
```

The beta-beta second partial derivative of the unit Dy vector is:

```    d2Dy      | -sin(alpha) * cos(beta) * cos(gamma) |
------  =  |  sin(alpha) * cos(beta) * sin(gamma) |
dbeta2     |      -sin(alpha) * sin(beta)         |
```

The beta-gamma second partial derivative of the unit Dy vector is:

```       d2Dy         | sin(alpha) * sin(beta) * sin(gamma) |
------------  =  | sin(alpha) * sin(beta) * cos(gamma) |
dbeta.dgamma     |                 0                   |
```

The gamma-gamma second partial derivative of the unit Dy vector is:

```    d2Dy       | -cos(alpha) * sin(gamma) - sin(alpha) * cos(beta) * cos(gamma) |
-------  =  | -cos(alpha) * cos(gamma) + sin(alpha) * cos(beta) * sin(gamma) |
dgamma2     |                             0                                  |
```

# Dz Hessian

The alpha-alpha second partial derivative of the unit Dz vector is:

```    d2Dz       | 0 |
-------  =  | 0 |
dalpha2     | 0 |
```

The alpha-beta second partial derivative of the unit Dz vector is:

```       d2Dz         | 0 |
------------  =  | 0 |
dalpha.dbeta     | 0 |
```

The alpha-gamma second partial derivative of the unit Dz vector is:

```        d2Dz         | 0 |
-------------  =  | 0 |
dalpha.dgamma     | 0 |
```

The beta-beta second partial derivative of the unit Dz vector is:

```    d2Dz      |  sin(beta) * cos(gamma) |
------  =  | -sin(beta) * sin(gamma) |
dbeta2     |        -cos(beta)       |
```

The beta-gamma second partial derivative of the unit Dz vector is:

```       d2Dz         | cos(beta) * sin(gamma) |
------------  =  | cos(beta) * cos(gamma) |
dbeta.dgamma     |           0            |
```

The gamma-gamma second partial derivative of the unit Dz vector is:

```    d2Dz       |  sin(beta) * cos(gamma) |
-------  =  | -sin(beta) * sin(gamma) |
dgamma2     |            0            |
```

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