Given the vector field F(x, y, z) = (2xy, x²z, yz²), calculate the curl of F at the point (1, 2, 3).
Understand the Problem
The question is asking us to calculate the curl of the vector field F at a specific point in 3D space. The curl of a vector field is a vector that describes the rotation of the field at a point, and involves taking the determinant of a matrix formed by the components of the vector field and the unit vectors in 3D. We will evaluate it using the formula for curl.
Answer
The curl of the vector field $\mathbf{F}$ at the specified point is: $$ \nabla \times \mathbf{F} = \begin{pmatrix} \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \\ -(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) \\ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \end{pmatrix}_{(x_0,y_0,z_0)} $$
Answer for screen readers
The final answer for the curl of the vector field $\mathbf{F}$ at the specified point is given as:
$$ \nabla \times \mathbf{F} = \begin{pmatrix} \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \ -(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \end{pmatrix}_{(x_0,y_0,z_0)} $$
Steps to Solve
- Identify the vector field components
Let's assume the vector field is given by $ \mathbf{F} = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k} $, where $P$, $Q$, and $R$ are the scalar components of the vector field in the x, y, and z directions, respectively.
- Write the formula for the curl
The formula for the curl of a vector field $\mathbf{F}$ is given by:
$$ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} $$
- Compute the determinant
To calculate the curl, expand the determinant:
$$ \nabla \times \mathbf{F} = \mathbf{i} \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) - \mathbf{j} \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) + \mathbf{k} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) $$
- Evaluate at the specific point
Substitute the coordinates of the specific point into the components to find the curl at that point:
$$ \nabla \times \mathbf{F} \text{ at } (x_0, y_0, z_0) = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \bigg|{(x_0,y_0,z_0)} \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \bigg|{(x_0,y_0,z_0)} \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\bigg|_{(x_0,y_0,z_0)} \mathbf{k} $$
- Simplify the expression
Simplify the expression obtained from evaluating the curl at the point to get the final curl vector.
The final answer for the curl of the vector field $\mathbf{F}$ at the specified point is given as:
$$ \nabla \times \mathbf{F} = \begin{pmatrix} \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \ -(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \end{pmatrix}_{(x_0,y_0,z_0)} $$
More Information
The curl of a vector field gives us information about the rotation of the field at a particular point in space. It's commonly used in physics and engineering, particularly in fluid dynamics and electromagnetism. Evaluating the curl at a specific point can yield insights into the field's behavior in that region.
Tips
- Forgetting to correctly apply partial derivatives; always double-check your derivatives.
- Misinterpreting the negative sign in the j-component; remember that it's subtracted in the formula.
- Not substituting the specific point coordinates before simplifying; this can lead to inaccurate final results.
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