Maths - Grad

Grad is short for gradient, it takes a scalar field as input and returns a vector field, for a 3 dimensional vector field it is defined as follows:

grad(s) = i ∂s + j ∂s + k ∂s
∂x ∂y ∂z

where,

Or in more conventional vector notation:

grad(s)=
∂s
∂x
∂s
∂y
∂s
∂z

The vector returned has a direction which shows the greatest rate of change and its magnitude |v| tells us what that rate of change is. If you apply a grad to, say the height of a hill, you will get a vector pointing directly uphill, the length of which tells you how steep the hill is. Perhaps that may not be a good example as the height is normally the third dimension in a position vector but here we are separating it out as a scalar on a 2 dimensional map. Another example might be temperature, which is a scalar quantity, which might vary over 3 dimensions. grad of this temperature field would show where the temperature is changing most rapidly and would give the direction of heat flow.

If n is a normal to the contour lines then

grad(s) = ∂s n
∂n

where:

n = normal vector

If the contour lines are closer together then grad(s) will be greater.

Vector Operator

The 'Del' operator represented by the 'Nabla' symbol∇is used to represent the vector operator, as follows:

∇= i + j + k
∂x ∂y ∂z

or in more conventional vector notation:

∇=
∂x
∂y
∂z

The del symbol does not have any meaning on its own, only when applied to a scalar field. Or a vector field as shown later.

∇s = grad(s)

Sum of Scalar Fields

Adding the scalar fields is equivalent to adding the grad functions:

grad(s1 + s2) = grad(s1) + grad(s2)

In the diagram below we have plotted the potential energy (scalar) field for:

  1. a single mass at x=-3
  2. a single mass at x=+3
  3. two masses at x=-3 and x=+3.

sumScalarFields

how this plot was produced.

As we can see the field for the two masses is the sum of the other two fields.

Identities

sum of scalar fields:

grad(s1 + s2) = grad(s1) + grad(s2)

product of scalar fields:

grad(s1s2) = s1 * grad(s2) + s2 * grad(s1)

grad of dot product of vector fields:

grad(v1v2) = (v2∇) v1 + (v1∇) v2 + v2 × curl(v1) +v1 × curl(v2)

where:

 



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see also:

Vectors

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