Can vector fields have gradients, and how are they calculated?

[wprince007]

The force on a magnetic dipole in a magnetic field is the dot product of the magnetic moment and the gradient of the field B, but gradients are operations done on scalar fields to produce vector fields. How does one calculate the gradient of a vector field if field gradients are only defined for scalar fields?

 

[ShayanJ]

The formula you're referring to is $\vec F=(\vec m \cdot \vec \nabla) \vec B$. Its not the dot product of $\vec m$ and the gradient of the magnetic field. Its expanded form is $\vec F=(m_x \frac{\partial}{\partial x}+m_y \frac{\partial}{\partial y}+m_z \frac{\partial}{\partial z})\vec B$.
You can also use the formula $\vec F= \vec\nabla(\vec m \cdot \vec B)$.

 

[wprince007]

Thanks Shyan...that clears things up. For some reason I wasn't notified of your response; I found this just now as I was trying to see if my query had gotten lost.