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Variables: Independent & Dependent

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Problem statement

The material derivative is written below in two different coordinate systems. One uses pressure as a vertical coordinate; the other uses height.

What are the dependent and independent variables? When taking partial derivatives in one variable, what other parameters are held constant? Write down the definition of w and ω in terms of differentials of the independent variables. What are the units of ω?

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Problem Answer (1)

The material derivative is written below in two different coordinate systems. One uses pressure as a vertical coordinate; the other uses height.

T, u, v, ω dependent variables

t, x, y, p independent variables

When taking partial derivatives in one independent variable, all other independent variables are held constant.

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Problem Answer (2)

The material derivative is written below in two different coordinate systems. One uses pressure as a vertical coordinate; the other uses height.

T, u, v, w dependent variables

t, x, y, z independent variables

When taking partial derivatives in one independent variable, all other independent variables are held constant.

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Problem Answer (3)

Write down the definition of w and ω in terms of differentials of the independent variables. What are the units of ω?