Gradient (B.Sc Physics)

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GRADIENT:

The gradient of a scalar field is a vector field & is represented by vector point function whose magnitude is equal to the maximum rate of change of scalar point function in a direction in which maximum rate of change occurs.

Let us consider two layers (scalar fields) P & Q, the values of scalar point functions are Φ & Φ+dΦ prospectively. So in moving from P to Q, the change in scalar field is dΦ.

In moving from A to B, the change in displacement is dr & change in scalar field is dΦ. So the rate of change of scalar field is dΦ/dr.

In moving from A to C, the change in displacement is dr' & change in scalar field is dΦ. So the rate of change of scalar field is dΦ/dr'.

In moving from A to D, the change in displacement is dr'' & change in scalar field is dΦ. So the rate of change of scalar field is dΦ/dr''.

Since dr<dr'<dr'' , So we get dΦ/dr>dΦ/dr'>dΦ/dr''.

So, the maximum rate of change is dΦ/dr & it is along the normal. Hence the maximum rate of change occurs along the normal & this is the direction of gradient.

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GRADIENT in Cartesian Co-ordinate System:

Let P(x,y,z) is a point situated in 3D-coordinate system. Inside this scalar field, let a particle is moving from P to Q. Then the change in scalar field is from Φ to Φ+dΦ for the points P & Q respectively. Due to this change let the increment along X,Y Z-axes are dx,dy & dz respectively.

So, dΦ/dx= Rate of change of scalar field along X-axis.

dΦ/dy= Rate of change of scalar field along Y-axis.

dΦ/dz= Rate of change of scalar field along Z-axis.

Let Φ=Φ(x,y,z)

The gradient is defined as

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PROPERTIES OF GRADIENT:

1. The line integral\of a vector field is independent of the path followed subject to the condition that the vector field is represented by the gradient of a scalar point function.