Question

Show that $$\frac{\delta \mathbf{E}_{\mathbf{y}, i}[u]}{\delta u}(\mathbf{x})=-\partial_{i} \delta(\mathbf{x}-\mathbf{y}) .$$

Short answer

The differentiation of the expectation of \(u\) with respect to \(u\) at \(\mathbf{x}\) is equal to the negative of the partial derivative of the Dirac delta function with respect to \(i\). This is proven using the properties of the Dirac delta function and the mechanics of differentiation under the integral sign associated with expectations.

Step-by-step solution

Differentiation of expectation

Start with the given expression \( \frac{\delta \mathbf{E}_{\mathbf{y}, i}[u]}{\delta u}(\mathbf{x}) \). The first step is to differentiate the expectation respect to the variable \( u \). Since it's an expectation, the derivative will carry across the integral that defines the expectation. Therefore, the expression will become \( -\mathbf{E}_{\mathbf{y}, i}[\partial_{i} u(\mathbf{x})] \).

Substitution of the variable

Next step is to substitute \( u(\mathbf{x}) = \delta(\mathbf{x}-\mathbf{y}) \), in the above expression, which gives \( -\mathbf{E}_{\mathbf{y}, i}[\partial_{i}\delta(\mathbf{x}-\mathbf{y})] \). The symbol \( \delta \) represents the Dirac Delta function.

Properties of Dirac Delta Function

Recall the property of the Dirac Delta function, which states that integrals of the spatial derivatives of a Dirac delta function over all space equals the negative of the value of the original function evaluated at the point where Dirac Delta function is not zero. Considering this property, the expectation of the integral transforms the equation to \( -\partial_{i}\delta(\mathbf{x}-\mathbf{y}) \).

Final Output

So, it proves that \( \frac{\delta \mathbf{E}_{\mathbf{y}, i}[u]}{\delta u}(\mathbf{x}) = -\partial_{i}\delta(\mathbf{x}-\mathbf{y}) \).

Key concepts

Dirac Delta function

The Dirac Delta function, denoted as \( \(\delta(x)\) \), is a fundamental concept in physics and engineering, especially in the realms of signal processing and quantum mechanics. At its core, it's not a function in the traditional sense but rather a 'generalized function' or 'distribution' that is used to model an infinitely sharp spike at a single point with an area of 1 under the spike. This means its value is zero everywhere except at the point \( x = 0 \), where it is theoretically infinite.

In mathematical terms, the Dirac Delta is defined by two key properties: first, \( \(\delta(x) = 0\) \) for \( x eq 0 \), and second, \( \int_{-\infty}^{\infty} \delta(x) dx = 1 \).

In the context of physics, it's often used to represent point charges, mass distributions, or initial conditions in a system. Its capacity to 'sift' out the value of a function at a particular point when integrated is a powerful tool in solving differential equations and in Fourier transforms.

Differentiation of expectation

Differentiation of expectation is a process that combines the operations of differentiation and taking an expectation in probability theory. Expectation, often represented by \( \(E[...]\) \), is a way of finding the average or expected value of a random variable. When differentiating an expectation, it's important to understand that if the expectation is with respect to a variable that differentiation is targeting, the differential operation can pass inside the integral representing the expectation.

So conceptually, when one differentiates the expected value of some function of a variable \( u \), it is akin to taking the derivative inside the integral that defines the expected value. This operation is crucial in stochastic calculus and shows up in the analysis of systems affected by randomness, like in quantum physics or financial mathematics.

Properties of Dirac Delta function

The Dirac Delta function has several unique properties that make it invaluable in mathematical physics. One central property is its 'sifting' characteristic, when it's integrated with another function: \( \int_{-\infty}^{\infty} f(x) \delta(x - a) dx = f(a) \) provided \( f \) is continuous at \( x = a \). This effectively evaluates the function \( f \) at the point \( a \).

Another essential property is related to differentiation, as highlighted in our original exercise. When a Dirac Delta function is differentiated, and then integrated over all space, it can yield the negative value of the function's derivative at the point where the delta function is centered. This reflects the idea that the Dirac Delta function, while being a theoretical construct, can mimic the behavior of derivatives of step functions in a distributional sense.

Additional properties include its scaling behavior and its role as an identity element under convolution, both of which are deeply connected to the theory of linear systems and Fourier analysis. Understanding these properties provides insight into the Delta function's bridging role between the discrete and continuous worlds.