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

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

Problem

Find the divergence 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 divergence of the vector field ā = P ī + Q j̄ + R k̄:

div \vec{a} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

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 divergence formula:

Why is this a scalar quantity?

Unlike the curl, the divergence is a scalar quantity. It is the sum of three partial derivatives, each of which is a scalar. Hence the result is also a scalar, not a vector.

\frac{\partial P}{\partial x} = 2 x y

\frac{\partial Q}{\partial y} = - 6 y^{2} z

\frac{\partial R}{\partial z} = 6 z x

Step 4

Substitute the found derivatives into the divergence formula:

div \vec{a} = 2 x y - 6 y^{2} z + 6 z x

Step 5

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

div \vec{a} (M) = 2 - 6 + 6 = 2

Answer

div ā(M) = 2.

Frequently asked questions

How do I find the divergence of a vector field?

The divergence of a vector field ā = P ī + Q j̄ + R k̄ is computed by: div ā = ∂P/∂x + ∂Q/∂y + ∂R/∂z. Find three partial derivatives and add them. The result is a scalar quantity.

What is the physical meaning of divergence?

Divergence shows how much a point is a source (div > 0) or a sink (div < 0) of the vector field. If div = 0, the field is called solenoidal — the flux through any closed surface is zero.

What is the difference between divergence and curl?

Divergence is a scalar quantity (characterises sources and sinks of the field). Curl is a vector quantity (characterises rotation of the field). Both may be non-zero for the same field, describing different properties.

How do I compute the divergence at a specific point?

First find the general expression for the divergence as a scalar function of coordinates (x, y, z). Then substitute the coordinates of the point M into this expression. In our case div ā(M) = 2.