Do Two Functions with Identical Gradient Vectors Differ? Exploring the Concept with a Practical Example

In the realm of calculus and vector calculus, it is a common belief that functions with the same gradient vectors are identical. However, this belief may not always hold true. This article delves into the concept by providing a practical example to demonstrate how two functions can have the same gradient vector yet still be different. We will also explore why this is important and how to prove such a statement.

Introduction to Gradient Vectors

A gradient vector, denoted as ?f(fx,fy,fz), represents the direction of the steepest increase of a scalar function f(x,y,z) and its magnitude indicates the rate of change in that direction. The gradient vector plays a crucial role in optimization, machine learning, and other mathematical fields.

Theoretical Insight

A fundamental theorem in vector calculus is that if two differentiable functions f(x,y,z) and g(x,y,z) have the same gradient vectors everywhere, then the functions differ by a constant. This implies that if two functions have different values at a particular point, their gradients must differ at that point.

Counterexample: Functions with Identical Gradient Vectors

Consider two simple functions in a two-dimensional space: f(x,y) and g(x,y).

Let f(x,y)ln(x)

and

Let g(x,y)2ln(x)

Both f and g are differentiable and their partial derivatives with respect to x are identical.

Compute the gradient of f and g:

Gradient of f(x,y)ln(x)

The partial derivative of f with respect to x is:

f x x 1 x

Since g is a scalar multiple of f, the gradient of g is also the same:

g x x 2 x

Both functions have the same gradient vector, but:

f (1)0 and g (1)0 f (2)ln(2) and g (2)2ln(2)

Thus, they are different even though they have the same gradient vector at every point x.

Implications and Practical Uses

The fact that two functions with identical gradient vectors can still differ has profound implications in fields such as machine learning, optimization, and geometry. It highlights the importance of considering the entire function and not just its gradient when making inferences or solving problems.

For instance, in machine learning, the landscape of the cost function is crucial for the training process. Knowing that a function can have the same gradient but different values can help in understanding the implications of local minima and maxima.

Proof of the Statement

To formally prove that two functions have the same gradient vectors but can still differ, consider the following:

Let f(x,y)ln(x) and g(x,y)2ln(x). We have:

f x x 1 x g x x

Since both functions have the same gradient with respect to x, they differ by a constant C such that:

g(x)f(x) C

Given f(1)0, we can determine that C0. However, at another point, say x2, we have f(2)ln(2) and g(2)2ln(2). This shows that the functions are different despite having the same gradient vector.

Conclusion

In summary, two functions with identical gradient vectors can still differ, provided that they have different values at some points. This concept is essential for understanding the behavior of functions in various mathematical and practical contexts. The example of f and g demonstrates this clearly. The knowledge of this property is vital, especially in optimization and machine learning, where the landscape of functions plays a crucial role.