Examples of sum rule in the following topics:
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- Kirchhoff's loop rule states that the sum of the emf values in any closed loop is equal to the sum of the potential drops in that loop.
- Kirchhoff's loop rule (otherwise known as Kirchhoff's voltage law (KVL), Kirchhoff's mesh rule, Kirchhoff's second law, or Kirchhoff's second rule) is a rule pertaining to circuits, and is based on the principle of conservation of energy.
- Mathematically, Kirchhoff's loop rule can be represented as the sum of voltages in a circuit, which is equated with zero:
- Kirchhoff's loop rule states that the sum of all the voltages around the loop is equal to zero: v1 + v2 + v3 - v4 = 0.
- An example of Kirchhoff's second rule where the sum of the changes in potential around a closed loop must be zero.
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- Kirchhoff's junction rule states that at any circuit junction, the sum of the currents flowing into and out of that junction are equal.
- Kirchhoff's junction rule, also known as Kirchhoff's current law (KCL), Kirchoff's first law, Kirchhoff's point rule, and Kirchhoff's nodal rule, is an application of the principle of conservation of electric charge.
- Kirchhoff's junction rule states that at any junction (node) in an electrical circuit, the sum of the currents flowing into that junction is equal to the sum of the currents flowing out of that junction.
- Thus, Kirchoff's junction rule can be stated mathematically as a sum of currents (I):
- We justify Kirchhoff's Rules from diarrhea and conservation of energy.
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- The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
- In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
- Some of the most basic rules are the following.
- Here the second term was computed using the chain rule and the third using the product rule.
- The flight of model rockets can be modeled using the product rule.
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- $\displaystyle f_{if} = \frac{B_{if}}{B_{if}^\mbox{classical}} = \frac{2 m }{3 \hbar^2 g_i e^2} \left ( E_f - E_i \right ) \sum |{\bf d}_{if}|^2$
- Here we have included the possibility that the lower state has a $g_f$-fold degeneracy and we have summed over the degenerate upper states.
- $\displaystyle f_{if} = \frac{2 m}{3 \hbar^2 g_i e^2} \left ( E_i - E_f \right ) \sum |{\bf d}_{if}|^2$
- There are several summation rules that restrict the values of the oscillator strengths,
- where the sum is over transitions that involve the outermost electrons.
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- This rule also applies to motion in a specific direction.
- We can easily extend this rule to the y-axis.
- In static systems, in which motion does not occur, the sum of the forces in all directions always equals zero.
- This rule also applies to rotational motion.
- We can represent this rule mathematically with the following equations:
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- Table 2.5 shows the probability distribution for the sum of two dice.
- A probability distribution is a list of the possible outcomes with corresponding probabilities that satisfies three rules:
- The probability distribution for the sum of two dice is shown in Table 2.5 and plotted in Figure 2.8.
- If the outcomes are numerical and discrete, it is usually (visually) convenient to make a bar plot that resembles a histogram, as in the case of the sum of two dice.
- 2.20: The probabilities of (a) do not sum to 1.
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- A series is the sum of the terms of a sequence.
- A series is, informally speaking, the sum of the terms of a sequence.
- The terms of the series are often produced according to a certain rule, such as by a formula or by an algorithm.
- The sequence of partial sums $\{S_k\}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence $\{a_n\}$ from $a_0$ to $a_k$:
- By definition, the series $\sum_{n=0}^{\infty} a_n$ converges to a limit $L$ if and only if the associated sequence of partial sums $\{S_k\}$ converges to $L$.
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- The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\, dx$.
- Popular methods use one of the Newton–Cotes formulas (such as midpoint rule or Simpson's rule) or Gaussian quadrature.
- The trapezoidal rule (also known as the trapezoid rule or trapezium rule) is a technique for approximating the definite integral $\int_{a}^{b} f(x)\,dx$.
- $\begin{aligned}\int_{a}^{b} f(x)\, dx &\approx \frac{h}{2} \sum_{k=1}^{N} \left( f(x_{k+1}) + f(x_{k}) \right) {} \\
&= \frac{b-a}{2N}(f(x_1) + 2f(x_2) + \cdots + 2f(x_N) + f(x_{N+1}))\end{aligned}$
- Use the trapezoidal rule to approximate the value of a definite integral
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- When there is unequal n, the sum of squares total is not equal to the sum of the sums of squares for all the other sources of variation.
- When confounded sums of squares are not apportioned to any source of variation, the sums of squares are called Type III sums of squares.
- When all confounded sums of squares are apportioned to sources of variation, the sums of squares are called Type I sums of squares.
- As you can see, with Type I sums of squares, the sum of all sums of squares is the total sum of squares.
- Even if the data analysis were to show a significant effect, it would not be valid to conclude that the treatment had an effect because a likely alternative explanation cannot be ruled out; namely, subjects who were willing to describe an embarrassing situation differed from those who were not.
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- Many statistical formulas involve summing numbers.
- If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.
- In this case the reader easily guesses the pattern; however, for more complicated patterns, one needs to be precise about the rule used to find successive terms.
- For example, the sum of $f(k)$ over all integers $k$ in the specified range can be written as: $\displaystyle \sum_{0\leq k }$
- The sum of $f(x)$ over all elements $x$ in the set $S$ can be written as: $\displaystyle \sum_{x\epsilon S}f(x)$