# How to differentiate by a vector ?

• dyn
In summary, the conversation discusses the possibility of differentiating a scalar (specifically a Lagrangian) by a vector and how this is defined. It is concluded that the vector derivative is a directional derivative in the direction of a unit vector and is often denoted as df/dA or D_Af. This notation has been used in older texts and is consistent with the Leibniz notation. It is suggested to include tensor indices when using this notation.
dyn
Is it possible to differentiate a scalar( in this case a Lagrangian) by a vector ?
If the Lagrangian is r.A ie. the scalar product of vectors r and A what is ∂L/∂r ? My notes say it is A but how ?
My notes also say that ∂L/∂r is the same as ∇L. Is this correct ? or is it sloppy notation ? or is it because the vector is treated as a generalised coordinate which acts as a scalar ?
Thanks

Sounds like they're just defining ##\frac{\partial L}{\partial\vec r}## to mean ##\left(\frac{\partial L}{\partial r_1} ,\frac{\partial L}{\partial r_2},\frac{\partial L}{\partial r_3}\right)##.

Fredrik's right - that's what they mean.

Thanks. Is it technically possible in precise terms to differentiate by a vector ? As far as I understand grad div and curl isn't differentiating by a vector.

For a function f that exists over a vector domain R the vector derivative is defined as:

df(R)/dA = lim [f(R + eps*A) - f(R)]/eps

Note that the vector A must be a unit vector here; the vector derivative is a directional derivative in the direction A.

You can easily prove that this can be calculated via:
A (dot) grad (f).

Do people use the notation df/dA for that? (I don't have a problem with it. I'm just not used to seeing it). I would use something like ##D_Af##.

I wouldn't call this "differentiating by a vector". It's just the directional derivative of f in the direction of the unit vector A. It's the ordinary derivative of the function ##t\mapsto f(x+tA)##, which is a function from ##\mathbb R## into ##\mathbb R##.

When all this occurs with the Euler-Lagrange equations is the vector just treated as a scalar ?

I'm not sure I even understand what it would mean to treat a vector as a scalar.

Fredrik said:
Do people use the notation df/dA for that? (I don't have a problem with it. I'm just not used to seeing it). I would use something like ##D_Af##.

I wouldn't call this "differentiating by a vector". It's just the directional derivative of f in the direction of the unit vector A. It's the ordinary derivative of the function ##t\mapsto f(x+tA)##, which is a function from ##\mathbb R## into ##\mathbb R##.

Yes, this notation is used, especially in older texts.

And yes, the directional derivative has been called "differentiating by a vector" ... the terminology is consistent with the Leibniz notation.

I like your notation better; it is more modern-looking. I'll use it next time and be more hip!

I don't think it's sloppy notation. It's just different notation. Although, it's probably better to include tensor indices when written that way.

## 1. What is a vector?

A vector is a mathematical object that has both magnitude (size) and direction. It is commonly represented by an arrow with a specific length and direction in a coordinate system.

## 2. How do you differentiate by a vector?

To differentiate by a vector means to find the rate of change of a function with respect to the direction of the vector. This is done by taking the dot product of the gradient of the function and the unit vector in the direction of the given vector.

## 3. What is the difference between differentiating by a vector and by a scalar?

When differentiating by a vector, we are considering the rate of change in a specific direction, whereas when differentiating by a scalar, we are finding the overall rate of change of a function.

## 4. What types of functions can be differentiated by a vector?

Any function that has multiple variables and is defined over a vector space can be differentiated by a vector. This includes functions such as velocity, acceleration, and force, which are commonly used in physics.

## 5. Why is vector differentiation important?

Vector differentiation allows us to analyze the rate of change of a function in a specific direction, which can be useful in many applications, such as in physics, engineering, and economics. It also helps us understand the relationship between different variables in a function and how they affect each other.

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