Gradient of ln(r) – Full Derivation & Formula

The gradient of the natural logarithm of r, denoted as ∇ln(r), is a fundamental operation in vector calculus with significant applications in electrostatics, potential fields, and fluid dynamics. Understanding its derivation is key for students in physics and engineering.

The Gradient of ln(r) Formula

We consider a position vector r in three-dimensional Cartesian coordinates:

r = xi + yj + zk

The magnitude of this vector, often called the radial distance, is given by:

r = |r| = √(x² + y² + z²)

The gradient of ln(r) is then defined as:

Where ∇ is the gradient operator, r is the radial distance, and r is the position vector.

Step-by-Step Derivation

To derive the formula ∇ln(r) = r/r², we use the chain rule for gradients. The chain rule states that for a function f(u), where u is itself a function of x, y, and z, the gradient is ∇f(u) = f'(u)∇u.

1. Define the function and its argument:
   Let f(r) = ln(r). Here, the outer function is the natural logarithm, and its argument is r.
2. Find the derivative of the outer function:
   The derivative of ln(r) with respect to r is f'(r) = 1/r.
3. Find the gradient of the inner function (∇r):
   First, it's easier to work with r². We have r² = x² + y² + z². Taking the gradient of both sides: ∇(r²) = ∇(x² + y² + z²).
   Using the chain rule, ∇(r²) = 2r∇r.
   Calculating the gradient of the right side: ∇(x² + y² + z²) = 2xi + 2yj + 2zk = 2r.
   So, we have 2r∇r = 2r. Solving for ∇r gives us:
   ∇r = r / r
4. Apply the chain rule:
   Now we combine our results using ∇ln(r) = f'(r)∇r.
   ∇ln(r) = (1/r) * (r/r) = r / r²

This completes the derivation, showing that the gradient of ln(r) is the position vector r divided by the square of its magnitude.

Applications

The result ∇ln(r) = r/r² is essential in many areas of physics and engineering:

• Potential Fields: In gravitational and electrostatic fields, the potential often varies with 1/r. Since the force is the negative gradient of the potential, this formula is used to find the force field, which varies as r/r².
• Electrostatics: It is fundamental to deriving the electric field from the electric potential of a point charge.
• Laplacian and Poisson Equations: The divergence of the gradient of ln(r) is related to the Dirac delta function, which is critical in solving Poisson's equation in various physical contexts.

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