Problem 14
Question
Determine whether the given vector field is conservative and/or incompressible. $$\left(2 x y, x^{2}-3 y^{2} z^{2}, 1-2 z y^{3}\right)$$
Step-by-Step Solution
Verified Answer
The given vector field is neither conservative nor incompressible.
1Step 1: Check if the vector field is conservative
To determine if a vector field is conservative, you need to check if it is path independent, which means the scalar potential should exist and the vector field should be its gradient. For a three-dimensional vector field, the cross derivative of its components should be equal. To verify this, calculate the cross derivative \(\partial F_{1} / \partial y\) and \(\partial F_{2} / \partial x\), \(\partial F_{2} / \partial z\) and \(\partial F_{3} / \partial y\), \(\partial F_{1} / \partial z\) and \(\partial F_{3} / \partial x\), where \(F = (F_{1}, F_{2}, F_{3})\) is the vector field. If these pairs of cross derivatives are not equal, the vector field is not conservative.
2Step 2: Perform calculations
The given vector field \(F = (2xy, x^2 - 3y^2z^2, 1 - 2zy^3)\). Calculate the cross derivatives: \(\partial F_{1} / \partial y = 2x\), \(\partial F_{2} / \partial x = 2x\), \(\partial F_{2} / \partial z = -6y^2z\), \(\partial F_{3} / \partial y = -6zy^2\), \(\partial F_{1} / \partial z = 0\), \(\partial F_{3} / \partial x = 0\). The cross derivatives are not equal, hence the vector field is not conservative.
3Step 3: Check if the vector field is incompressible
A vector field is incompressible if its divergence equals to zero. The divergence of a vector field \(F\) in three dimensions is given by \(\nabla \cdot F = \partial F_{1} / \partial x + \partial F_{2} / \partial y + \partial F_{3} / \partial z\). Thus, calculate the divergence with these equations.
4Step 4: Perform calculations
For the given vector field \(F = (2xy, x^2 - 3y^2z^2, 1 - 2zy^3)\), calculate the divergence: \(\nabla \cdot F = \partial F_{1} / \partial x + \partial F_{2} / \partial y + \partial F_{3} / \partial z = 2y + 2x - 6zy^2 = 2(x+y) - 6zy^2\). As the divergence is not equal to zero, the vector field is compressible.
Key Concepts
Path IndependenceScalar PotentialDivergence of a Vector FieldCross Derivatives
Path Independence
Path independence is an intriguing property of certain vector fields that ties deeply into the concept of work done by a force. In the physical world, if a force field (like a gravitational or electric field) is path independent, the work done to move an object from one point to another solely depends on these points and not on the path taken between them. Mathematically, this is true for conservative vector fields.
In the context of the given exercise, to test for path independence, we would check if the vector field is the gradient of some scalar potential function. This is equivalent to checking whether the cross derivatives of the field's components are equal. For the given vector field \(2xy, x^2 - 3y^2z^2, 1 - 2zy^3\), the calculation of cross derivatives indicated that the vector field is not conservative, thereby implying that the field is path dependent. If the field were conservative, this would allow us to find a scalar potential function, essentially simplifying the process of understanding the field's behavior.
In the context of the given exercise, to test for path independence, we would check if the vector field is the gradient of some scalar potential function. This is equivalent to checking whether the cross derivatives of the field's components are equal. For the given vector field \(2xy, x^2 - 3y^2z^2, 1 - 2zy^3\), the calculation of cross derivatives indicated that the vector field is not conservative, thereby implying that the field is path dependent. If the field were conservative, this would allow us to find a scalar potential function, essentially simplifying the process of understanding the field's behavior.
Scalar Potential
The scalar potential is a foundational concept that captures the potential energy per unit charge or mass at a point within a field. In a vector field context, a scalar potential \(\phi\) is a function from which the vector field can be derived as its gradient, represented as \(abla \phi\). This means that every conservative vector field has an associated scalar potential.
In our textbook example, the absence of a scalar potential is noted due to the unequal cross derivatives of the vector field's components. If the scalar potential had existed, it would have provided invaluable insights into the field's behavior, because the potential difference between two points would reveal the work done against the field.
In our textbook example, the absence of a scalar potential is noted due to the unequal cross derivatives of the vector field's components. If the scalar potential had existed, it would have provided invaluable insights into the field's behavior, because the potential difference between two points would reveal the work done against the field.
Divergence of a Vector Field
Divergence is a measure of a vector field's tendency to originate from or converge into a point, indicating whether a field is 'spreading out' or 'contracting in' at a point. For vector fields in three-dimensional space, we define divergence as the dot product of the del operator \(abla\) with the vector field, \(abla \cdot F\).
Regarding our given vector field \(2xy, x^2 - 3y^2z^2, 1 - 2zy^3\), we calculated the divergence and found it to be unequal to zero. Therefore, we concluded that the vector field is compressible, meaning the density of the field lines changes over space.
Regarding our given vector field \(2xy, x^2 - 3y^2z^2, 1 - 2zy^3\), we calculated the divergence and found it to be unequal to zero. Therefore, we concluded that the vector field is compressible, meaning the density of the field lines changes over space.
Cross Derivatives
Cross derivatives are partial derivatives that involve taking the derivative of one component of a vector field with respect to a variable of another component. They are crucial in determining a vector field's conservativeness. For a three-dimensional vector field \(F_1, F_2, F_3\), we check if \(\partial F_{1} / \partial y = \partial F_{2} / \partial x\), \(\partial F_{2} / \partial z = \partial F_{3} / \partial y\), and \(\partial F_{1} / \partial z = \partial F_{3} / \partial x\).
In the exercise, calculating these cross derivatives showed that they were not equal, clearly indicating that the given vector field could not be described purely by a scalar potential and hence was not conservative. If they were equal, it would suggest that there exists a function whose gradient would reproduce the vector field fully, simplifying many calculations and further analysis.
In the exercise, calculating these cross derivatives showed that they were not equal, clearly indicating that the given vector field could not be described purely by a scalar potential and hence was not conservative. If they were equal, it would suggest that there exists a function whose gradient would reproduce the vector field fully, simplifying many calculations and further analysis.
Other exercises in this chapter
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