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Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous first-order partial derivatives, then the following is true. āˆ‚P āˆ‚y = āˆ‚Q āˆ‚x āˆ‚P āˆ‚z = āˆ‚R āˆ‚x āˆ‚Q āˆ‚z = āˆ‚R āˆ‚y . Since F is conservative, there exists a function f such that F = āˆ‡f, that is, P, Q, and R are defined as follows. (Enter your answers in the form fx, fy, fz.)

Respuesta :

abidemiokin

Answer: The field F has a continuous partial derivative on R.

Step-by-step explanation:

For the field F has a continuous partial derivative on R, fxy must be equal to fyx and since our field F is āˆ‡f,

āˆ‡f = fxi + fyj + fzk.

Comparing the field F to āˆ‡f since they at equal, P = fx, Q = fy and R = fz

Since P is fx therefore;

āˆ‚P āˆ‚y = āˆ‚ āˆ‚y( āˆ‚f āˆ‚x) = āˆ‚2f āˆ‚yāˆ‚x

Similarly,

Since Q is fy therefore;

āˆ‚Q āˆ‚x = āˆ‚ āˆ‚x( āˆ‚f āˆ‚y) = āˆ‚2f āˆ‚xāˆ‚y

Which shows that āˆ‚P āˆ‚y = āˆ‚Q āˆ‚x

The same is also true for the remaining conditions given

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