It is just one component of the fully completed equation.
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ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.
Yes, it is.
partial of u with respect to x = partial of v with respect to y partial of u with respect to y = -1*partial of v with respect to x
Some partial differential equations do not have analytical solutions. These can only be solved numerically.
partial vx w/ respect to x + partial vy w/ respect to y + partial vz w/ respect to z = 0