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If the vector field F(x, y, z) = (ayº z2, bxy? x2, 2xyz) is conservative then a+b= y, = z2

User Ahmer Khan
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Answer:

To determine if the vector field F(x, y, z) = (ayº z2, bxy? x2, 2xyz) is conservative, we can analyze its properties by considering the conditions for conservative vector fields.

For a vector field F to be conservative, it must satisfy the condition that the curl of F is equal to zero. The curl of a vector field F = (F1, F2, F3) is given by the following expression:

curl(F) = (∂F3/∂y - ∂F2/∂z, ∂F1/∂z - ∂F3/∂x, ∂F2/∂x - ∂F1/∂y)

Let's compute the curl of F:

∂F3/∂y = 2xz

∂F2/∂z = 0

∂F1/∂z = 0

∂F3/∂x = 2yz

∂F1/∂y = az2

∂F2/∂x = by

By substituting the partial derivatives into the curl expression, we can calculate:

curl(F) = (2xz - 0, 0 - 2yz, by - az2) = (2xz, -2yz, by - az2)

To satisfy the condition for a conservative vector field, the curl of F must be equal to zero:

2xz = 0

-2yz = 0

by - az2 = 0

From the first equation, we have x = 0 or z = 0, or both. From the second equation, we have y = 0 or z = 0, or both. However, since we are looking for a condition that involves a and b, we can ignore the equations involving x and y.

Now, let's consider the third equation, by - az2 = 0. Since this equation must hold for all values of x, y, and z, we can conclude that:

by - az2 = 0 => b = az2

We now have an expression for b in terms of a.

From the condition for a and b given in the question, a + b = y, and a + az2 = y. By comparing the expressions, we can see that:

a + az2 = y

a(1 + z2) = y

We can conclude that for the vector field F(x, y, z) to be conservative with these conditions, a and b must satisfy the equations b = az2 and a(1 + z2) = y.

Please note that without specific values or constraints for y and z, we cannot determine the specific values of a and b. However, we have derived the relationships between a and b based on the conditions provided.

User Martin M J
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