Curl of a vector field at M(1;1;1)

This problem asks to find the curl of a vector field at a given point.

Problem

Find the curl of the vector field:

\vec{a} = x^{2} y \vec{i} - 2 y^{3} z \vec{j} + 3 z^{2} x \vec{k}, M (1; 1; 1)

Solution

Step 1

Write the formula for the curl of the vector field ā = P ī + Q j̄ + R k̄:

rot \vec{a} = (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}) \vec{i} - (\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) \vec{j} + (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) \vec{k}

Step 2

Write out the components of the vector field:

P = x^{2} y, Q = - 2 y^{3} z, R = 3 z^{2} x

Step 3

Compute the partial derivatives appearing in the curl formula:

Why are some derivatives zero?

A derivative is zero when the corresponding variable does not appear in the component of the field. For example, ∂R/∂y = 0 because R = 3z²x does not depend on y. This is typical for polynomial fields and significantly simplifies the computation.

\frac{\partial R}{\partial y} = 0, \frac{\partial Q}{\partial z} = - 2 y^{3}

\frac{\partial R}{\partial x} = 3 z^{2}, \frac{\partial P}{\partial z} = 0

\frac{\partial Q}{\partial x} = 0, \frac{\partial P}{\partial y} = x^{2}

Step 4

Substitute the found derivatives into the curl formula:

rot \vec{a} = (0 - (- 2 y^{3})) \vec{i} - (3 z^{2} - 0) \vec{j} + (0 - x^{2}) \vec{k}

= 2 y^{3} \vec{i} - 3 z^{2} \vec{j} - x^{2} \vec{k}

Step 5

Substitute the coordinates of the point M(1; 1; 1) into the expression for the curl:

rot \vec{a} (M) = 2 \vec{i} - 3 \vec{j} - \vec{k} = (2; - 3; - 1)

Answer

rot ā(M) = (2; −3; −1).

Frequently asked questions

How do I find the curl of a vector field?

The curl of a vector field ā = P ī + Q j̄ + R k̄ is computed by: rot ā = (∂R/∂y − ∂Q/∂z) ī − (∂R/∂x − ∂P/∂z) j̄ + (∂Q/∂x − ∂P/∂y) k̄. Find six partial derivatives and substitute them into this formula.

What is the geometric meaning of the curl?

The curl (or vortex) of a vector field characterises the rotational component of the field at a given point. The curl vector is directed along the axis of rotation, and its magnitude is proportional to the angular velocity. If the curl is zero, the field is called irrotational (potential).

What is the difference between curl and divergence?

The curl is a vector quantity characterising the rotation of the field. The divergence is a scalar quantity characterising the presence of sources (sinks) of the field. Both quantities may be non-zero for the same field, but they describe different properties.

How do I compute the curl at a specific point?

First find the general expression for the curl as a vector function of coordinates (x, y, z). Then substitute the coordinates of the point M into this expression. In our case rot ā(M) = (2; −3; −1).