Posts Tagged ‘gradient’

Vectors in 3 dimensions

In this blog, I explained some of the basics of vectors. I only dealt with 2-dimensional vectors, but of course we live in a 3-dimensional world. So, to fully specify a vector in 3-dimensions we need to add a third component. Usually, but not always, we can use the x, y and z-components of the vector, and so can write any vector in terms of its x, y and z-components.

For example, let us suppose we have some force \vec{F} which is in 3 dimensions, then we can write it in terms of its Cartesian (x,y,z) components as \vec{F} = F_{x} \; \hat{x}  + F_{y} \; \hat{y} + F_{z} \; \hat{z}, where \hat{x}, \hat{y}, \hat{z} are the unit vectors in the (x,y,z) directions respectively, and F_{x}, F_{y} and F_{z} are the x, y and z-components respectively (in this figure the unit vectors \hat{x}, \hat{y} and \hat{z} are labelled \hat{i}, \hat{j} and \hat{k}, which is an alternative nomenclature ).

The 3-D force F can be broken down into its 3 components in the x, y and z-directions.

The 3-D force F can be broken down into its 3 components in the x, y and z-directions.

Sometimes, we may be interested in how much the vector is changing with position. A good example of this might be the force on a charged particle produced by a bar magnet. We have all seen the so-called “field pattern” produced by the magnetic field of a bar magnet. We sprinkle iron filings on a piece of card, put the bar magnet under the card, and the iron filings align along the field lines.

The field lines of a bar magnet.

The field lines of a bar magnet.

We might be interested in how much the force due to this magnetic field varies with position. To make things easier, if we only consider the change in the strength of the force on the surface of the card, then we can ignore any change in the vertical dimension, which we will call the z-dimension.

How much a quantity changes with position can be determined by taking the derivative with respect to x, with respect to y and with respect to z, our three spatial directions.

The vector differential operator

The vector differential operator \nabla (often called del) is the mathematical operator which determines how much vectors change in each of their 3 spatial components. Mathematically it can be written as the partial derivative in each of the 3 spatial dimesions, so usually

\nabla = \hat{x} \frac{\partial }{\partial x} + \hat{y} \frac{\partial }{\partial y} + \hat{z} \frac{\partial }{\partial z}

if we are using Cartesian coordinates. The partial derivative \partial is the derivative with respect to one of the variables when the other variables are kept constant. For example, suppose we had a force field which could be described by the following (completely arbitrary) equation

\vec{F} = (2x)  \hat{x} + (3yz) \hat{y} + (4xy) \hat{z}

What would be the partial derivative with respect to x of this force?

\frac{\partial}{\partial x}\vec{F} = \frac{\partial}{\partial x}(2x + 3yz + 4xy)= 2 +4y

The partial derivate with respect to y would be

\frac{\partial}{\partial y}\vec{F} = \frac{\partial}{\partial y}(2x + 3yz +4xy) = 3z + 4x

and the partial derivate with respect to z would be

\frac{\partial}{\partial z}\vec{F} = \frac{\partial}{\partial z}(2x + 3yz + 4xy) = 3y.

So, finally, we can write \boxed{ \nabla \vec{F} =  (2+4y)\hat{x} + (3z + 4x)\hat{y} + (3y)\hat{z} }.

The term we use for a quantity like \nabla \vec{F} is the gradient (or grad) of the vector. This word derives from the fact that, in two dimensions, if we have y=f(x), then the gradient of the function (the slope of the function) is just given by \frac{d}{dx}f(x). So the grad is just the three (or more) dimensional equivalent of taking the derivative of a two dimensional function like y=f(x).

As I will explain in a future blog, the vector differential operator \nabla is a very powerful and important operator in mathematics and physics. It forms the basis of vector calculus and can be combined in two other ways with a vector to derive the divergence of the vector and its curl.

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