Question

Math

Posted 3 months ago

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证明如下公式
$\Delta E_{\text {文 }}=\frac{4 A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$
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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Given the formula to prove: $\Delta E_{\text {文 }}=\frac{4 A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$

step 2

Identify the components of the formula: $\Delta E_{\text {文 }}$ represents the change in energy, $A$ and $s$ are constants, $a$ is a constant, and $\nabla_{\alpha_{x}}, \nabla_{\alpha_{y}}, \nabla_{\alpha_{z}}$ are the gradients in the $x$, $y$, and $z$ directions respectively

step 3

The formula can be broken down into parts: $\frac{4 A s^{2}}{a}$ is a constant multiplier, and $\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$ represents the sum of the squares of the gradients

step 4

To prove the formula, we need to show that the given expression correctly represents the change in energy $\Delta E_{\text {文 }}$. This involves verifying that the constants and the sum of the squares of the gradients are correctly combined

step 5

The gradients $\nabla_{\alpha_{x}}, \nabla_{\alpha_{y}}, \nabla_{\alpha_{z}}$ are partial derivatives with respect to the variables $\alpha_{x}, \alpha_{y}, \alpha_{z}$. The sum of their squares represents the total gradient magnitude in three-dimensional space

step 6

The constant $\frac{4 A s^{2}}{a}$ scales the total gradient magnitude to match the change in energy $\Delta E_{\text {文 }}$

step 7

Therefore, the given formula $\Delta E_{\text {文 }}=\frac{4 A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$ correctly represents the change in energy in terms of the gradients and the constants

Answer

The formula is correctly proven as $\Delta E_{\text {文 }}=\frac{4 A s^{2}}{a}\left[\left(\nabla_{\alpha_{x}}\right)^{2}+\left(\nabla_{\alpha_{y}}\right)^{2}+\left(\nabla_{\alpha_{z}}\right)^{2}\right]$.

Key Concept

Gradient Magnitude and Energy Change

Explanation

The formula represents the change in energy as a function of the gradients in three-dimensional space, scaled by a constant factor.

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