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For instance, the scalar time derivative of u is expressed as,

(3.1.3)

When u is replaced with v and w in the above equation, then the corresponding expressions would be,

(3.1.4)

Now, summing them into a vector quantity, one may write Eq. (3.1.2) in compact form as,

(3.1.5)

In the above equation, is called as “local acceleration” and the second part i.e. is called a “convective acceleration”. The total time derivative is called as “substantial/material” derivative. This field concept can be applied to any variable (vector or scalar). For example, one may write the total derivative for pressure and temperature field as,

(3.1.6)