Derivative of An Inner Product

In summary, the conversation discusses the application of the chain rule for inner products, specifically if it holds true for derivatives and abstract inner products. The participants also mention the proof for the chain rule and its requirements.
  • #1
brydustin
205
0
I am trying to take the derivative of an
inner product (in the most general sense
over L^2), and was curious if the
derivative follows the "chain rule" for
inner products.

i.e. Does D_y(<f,g>) = <D_y(f),g> + <f,D_y(g)>
where D_y is the partial derivative w.r.t. y.

So for example, IT IS TRUE that if f=x*y and g=sin(x*y)
and the inner product <f,g> = Integral(f#g, w.r.t. x,-Pi,+Pi), f# = complex conj. of f.
then the equality holds.
In other words, differentiating w.r.t. y and integrating w.r.t x the forumla holds.
It seems more trivial if the variable which is being differentiated & integrated is the same.
But is it true in general?
What if we are differentiating more abstract inner products (i.e. not necessarily integration).
 
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  • #2
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Related to Derivative of An Inner Product

1. What is the definition of the derivative of an inner product?

The derivative of an inner product is a mathematical concept that represents the rate of change of an inner product between two vectors with respect to a variable. It is denoted by d(u,v), where u and v are the two vectors in the inner product.

2. How is the derivative of an inner product calculated?

The derivative of an inner product is calculated using the properties of inner products and the rules of differentiation. It involves taking the derivative of each vector in the inner product separately and then combining them using the properties of inner products.

3. What is the significance of the derivative of an inner product?

The derivative of an inner product has several applications in mathematics and physics. It can be used to find the direction of steepest ascent or descent in a multivariable function, as well as to optimize functions and solve optimization problems. It is also used in physics to calculate rates of change in physical systems.

4. Can the derivative of an inner product be negative?

Yes, the derivative of an inner product can be negative. The value of the derivative depends on the vectors in the inner product and the variable with respect to which it is being calculated. If the vectors are changing in opposite directions, the derivative can be negative.

5. How is the derivative of an inner product related to the gradient?

The derivative of an inner product is closely related to the gradient of a function. In fact, the gradient is the vector that results from taking the derivative of a function with respect to each of its variables. Therefore, the derivative of an inner product can be thought of as a generalization of the gradient to inner products between vectors.

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