Assumption 3: Under the assumption of infinitesimal displacements, the deflections  are small. Therefore the Jacobian can be assumed to be linearly dependent on them by neglecting the higher powers of , and also their product. Therefore, the jacobian can be expressed as
Substituting in Equation 39, we get
Simplifying further we get
Using Equations 36 and 37 for  , and integrating over the dimensions of the growth chamber, we get
Equation 44 is the governing equation of the shadowgraph process under the set of linearizing approximation 1-3. In concise from the above equation can be rewritten as
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