Relationship between differential and the operator

In summary: This is not a true statement. In reality, df = d/dx * dx, meaning that the change [of a function f] is equal to the derivative operator with respect to x multiplied by the change in x. This is known as the limit definition of the derivative. In the case of multivariable functions, the differential operator can be thought of as a unit vector, and the differential of x is the standard basis covector. In this basis, the derivative operator sends a function to its coefficients in terms of dx. However, it is not possible to assign a numerical value to dx and d/dx as they are both operators. It is important to understand the limit definition of the derivative when dealing with cases like this.
  • #1
wumple
60
0
Is there some sort of relationship between the differential

[tex]dx[/tex]

and the differential operator which means to take the derivative [tex]d/dx[/tex]

if x is a dependent variable? My prof said that [tex] dx * d/dx = 1[/tex] but that doesn't seem to work out in the case I'm looking at, so I must be missing something.
 
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  • #2
[itex]dx \frac{d}{dx}[/itex] is just an operator multiplied by a differential. If you apply it to a function f(x), it'll give the differential df, however. So you could call it the function differential operator, I suppose.
 
  • #3
So then can you take [tex]dx = 1/(d/dx)[/tex]? As in the differential is one over the operator? Or does that not work? I was having trouble making that work out in one of my problems.
 
  • #4
No, that wouldn't work. I really don't believe it's possible to say that dx d/dx = 1, anyway. If you're really trying to assign it a value, it would be just d, or a differential operator. That is, you're basically saying "the derivative operator with respect to a variable, multiplied by the differential of that variable, is the differential operator".
 
  • #5
You might want to revert back to the limit definition for the derivative for understanding cases like this.
 
  • #6
On a multivariable function, [itex]\frac{\partial}{\partial x}[/itex] and [itex]\partial x[/itex] can be thought of as unit vectors. I am still not totally comfortable with the idea, but perhaps this is what your prof was getting at?
 
  • #7
have you had linear algebra? df is a field of (co)vectors, and dx is a field of standard basis covectors. In this basis, d/dx is the operator that sends f to its field of coefficients in terms of the basis dx. I.e. at every point p, df(p) = df/dx(p) dx.

I.e. at every point df/dx(p) is a number, and at every point df(p) is a covector.

At every point dx(p) is also a covector, and we have at every point, that the number df/dx(p) times the covector dx(p), equals the covector df(p).

so df/dx . dx = df, an equation that holds at every point p.for instance if f(x) = x^2, then df(p) = 2p. dx(p) = df/dx(p) dx(p).
 
  • #8
wumple said:
Is there some sort of relationship between the differential

[tex]dx[/tex]

and the differential operator which means to take the derivative [tex]d/dx[/tex]

if x is a dependent variable? My prof said that [tex] dx * d/dx = 1[/tex] but that doesn't seem to work out in the case I'm looking at, so I must be missing something.

Your professor is wrong. [tex] dx * d/dx = d[/tex]. In words, the change [of something] per change in x times the change in x is the change [in the thing].
 

1. What is the relationship between differential and the operator?

The relationship between differential and the operator is that the differential operator is a mathematical operation that is used to calculate derivatives, which are essentially the rate of change of a function with respect to its variables. The differential operator is typically represented by the symbol "d/dx", where "d" represents differentiation and "dx" represents the variable with respect to which the function is being differentiated.

2. How is the differential operator used in calculus?

The differential operator is used in calculus to find the derivative of a function. It is a shorthand notation that allows us to easily find the derivative of a function without having to write out the limit definition every time. The differential operator is also used in solving differential equations, which are equations that involve derivatives of a function.

3. What are the different types of differential operators?

There are several types of differential operators, including the first-order derivative operator, the second-order derivative operator, and the higher-order derivative operator. These operators are used to calculate the first, second, and nth derivatives of a function, respectively. There are also partial derivative operators, which are used to calculate the derivatives of multivariable functions.

4. How does the differential operator relate to other mathematical operations?

The differential operator is closely related to other mathematical operations, such as integration and summation. In fact, the derivative and the integral are inverse operations of each other. This means that the integral of a function is the antiderivative of that function, and the derivative of an integral is the original function. Additionally, the differential operator can also be used to calculate the gradient of a function, which is a vector that points in the direction of the greatest rate of change of that function.

5. What are some real-world applications of the differential operator?

The differential operator has many practical applications in various fields, including physics, engineering, economics, and more. For example, it is used in physics to calculate the velocity and acceleration of objects in motion, and in economics to model and predict changes in supply and demand. The differential operator is also used in engineering to analyze and design systems that involve rates of change, such as control systems and circuits.

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