Calculating Gradient of ln(r) in 3D Space

  • Thread starter brentd49
  • Start date
  • Tags
    Derivative
In summary, the formula for calculating the gradient of ln(r) in 3D space is ∇ln(r) = (1/r)∇r = (1/r)(∂r/∂x, ∂r/∂y, ∂r/∂z). The gradient represents the rate of change or slope of the natural logarithm of the distance from the origin in 3D space. The partial derivatives in the formula can be calculated using basic calculus rules. The gradient can be negative, indicating a decrease in ln(r) as we move in a certain direction. It has various real-world applications in physics, engineering, and computer graphics.
  • #1
brentd49
74
0
So, the question is: find the gradient of f=ln(r), where r=sqrt(x^2+y^2+z^2)^(1/2).

For the partial with respect to x, I use the chain rule: df/dr*dr/dx.

df/dr=1/r
dr/dx=(1/2)*(2x)=x

Which would give df/dx = x/r.

But the book gets x/r^2

Where does the extra 1/r come from?
 
Physics news on Phys.org
  • #2
brentd49 said:
So, the question is: find the gradient of f=ln(r), where r=sqrt(x^2+y^2+z^2)^(1/2).
...
dr/dx=(1/2)*(2x)=x
...
This isn't right. Remember to use the chain rule here as well.
 
  • #3
hah. of course. thanks
 

FAQ: Calculating Gradient of ln(r) in 3D Space

1. What is the formula for calculating the gradient of ln(r) in 3D space?

The formula for calculating the gradient of ln(r) in 3D space is:
∇ln(r) = (1/r)∇r = (1/r)(∂r/∂x, ∂r/∂y, ∂r/∂z)

2. What does the gradient of ln(r) represent in 3D space?

The gradient of ln(r) represents the rate of change or the slope of the natural logarithm of the distance from the origin in 3D space. It tells us how quickly the value of ln(r) changes as we move in the x, y, and z directions.

3. How do you calculate the partial derivatives in the formula for the gradient of ln(r) in 3D space?

The partial derivatives in the formula for the gradient of ln(r) in 3D space can be calculated using basic calculus rules. The derivative of ln(r) with respect to x is ∂ln(r)/∂x = 1/r * ∂r/∂x. Similarly, the derivatives with respect to y and z can be calculated in the same way.

4. Can the gradient of ln(r) in 3D space be negative?

Yes, the gradient of ln(r) in 3D space can be negative. This indicates that the value of ln(r) decreases as we move in that particular direction. However, the magnitude of the gradient is more important than its sign, as it tells us about the rate of change or slope of ln(r).

5. How is the gradient of ln(r) used in real-world applications?

The gradient of ln(r) has many real-world applications, such as in physics, engineering, and computer graphics. It is used to calculate the electric field and gravitational potential in 3D space, and also to create 3D terrain maps in computer graphics. In physics, it is used to calculate the force and torque on an object in a 3D space.

Similar threads

Can anyone please verify/confirm these derivatives?
Replies
5
Views
1K
Integration problem using inscribed rectangles
Replies
14
Views
1K
Finding the partial derivative from the given information
Replies
2
Views
1K
For this Partial Derivative -- Why are different results obtained?
Replies
23
Views
2K
Computing line integral using Stokes' theorem
Replies
8
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
A question about the derivation of upper bound integrals
Replies
23
Views
2K
Lagrange multipliers understanding
Replies
8
Views
1K
Simmons 7.10 & 7.11: Find Curves Intersecting at Angle pi/4
Replies
1
Views
1K
Cauchy Riemann complex function real and imaginary parts
Replies
19
Views
1K

Hot Threads

Recent Insights

Back
Top