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Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.
ssh said:Q. Show that f(z)dz defined in a region is exact if and only if f(z) has a primitive.
An exact differential is a type of differential that represents the total change in a function with respect to its independent variables. It is exact because it does not depend on the path taken to reach a particular point, only the initial and final values.
To determine if f(z)dz is exact, you can use the partial derivative test. This involves taking the partial derivatives of the function with respect to each of its independent variables and checking if they are equal. If they are equal, then the function is exact.
No, f(z)dz cannot be exact for all values of z. It is only exact for a specific set of values, typically within a certain domain or range. Outside of this range, the function may not be exact.
If f(z)dz is exact, it means that the function has a well-defined total change and can be integrated exactly. This can make solving certain mathematical problems much easier and more accurate.
No, f(z)dz can only be exact for a conservative vector field. This means that the vector field satisfies the property of conservative vector fields, where the line integral of the field is independent of the path taken. If the vector field is not conservative, then f(z)dz will not be exact.