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Idius Land! » Blog Archive » And Now, A Two-Point Cubic Spline

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Laquo; Quadratic Spline Interpolation. And Now, A Two-Point Cubic Spline. The simple quadratic spline discussed my previous post seems to be sufficient for interpolating particle positions between two GADGET-2 snapshots for the purposes of making animations, but it's also. To use a cubic spline for the task. The advantages of using a cubic spline are (1) it can be more accurate—especially in cases where the acceleration changes sign. And solve the system of equations for. In terms of these definitions,.

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Idius Land! » Completing the Square

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The method of "completing the square" is sometimes useful in simplifying expressions. To see how it works, let's write out what happens when we square the expression. Notice the pattern on the right side: it's the square of each term plus twice the mixed product. It always works this way. You can use the pattern to force an expression to contain a perfect square. For instance. Can be re-written to make use of the pattern if you factor out. Correspond to one another,. Enables you to write.

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Idius Land! » The Product Rule

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In calculus textbooks, the product rule for differentiation is often presented as a theorem. The authors then prove the theorem using the definition of the derivative. Other authors choose to simply write out the steps of the proof and present the product rule as a useful identity without motivating the mathematical steps (see MathWorld. Suppose the product we are differentiating is the area of a rectangle,. If the width and length of the rectangle are. Further suppose that, in a time interval.

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Idius Land! » Euler’s Equation

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In this section, we consider the important complex function,. With its definition,. Part is easy to understand, since it is the ordinary exponential function with a real argument, but what about the second exponential on the right hand side of the equation? What does it mean for an exponent to be imaginary? To find out, we can expand the term. Grouping the real and imaginary parts,. This is known as. This is known as. It is significant because it relates the fundamental numbers,.

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Idius Land! » Complex Numbers

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Represents a positive, real number. In order to solve equations of the form. We would naturally write. But how exactly do we compute the square root of a negative number? As long as we use real numbers, there is no way to write the value of. In this case. In order to remedy this, mathematicians invented (or discovered) a type of number called "imaginary numbers." The idea is simple; we can write the square root as. For example, solving. Another problem arises if we try to solve the quadratic. Until this ...

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Idius Land! » Integration by Parts

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One often encounters integrands which can be written as the product of two functions. Sometimes the integral can be solved using the "u-substitution" method which is essentially the inverse of the chain rule. In other cases, integration by parts may work. Integration by parts is the inverse of the product rule of differentiation. Recall that the product rule states. If we integrate this equation, we get. Now if we rearrange a bit, this becomes. This result is the what is known as "integration by parts".

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Idius Land! » Trigonometric Functions

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It is possible to write the basic trig functions in terms of complex exponentials. Cosine and sine are just the real and imaginary parts of the complex exponential. The tangent is then the ratio. The secant, cosecant, and cotangent are just the reciprocals. The angle addition formulas can be derived easily using the complex exponential forms. First let's look at angle addition formulas for sine and cosine. The real and imaginary parts give the results. The double angle formulas can be found by setting.

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Idius Land! » Inverse Trig Derivatives

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This is a nice trick for calculating the derivatives of inverse trig functions. I'll demonstrate the process for the derivative of. The rest of the inverse trig derivatives can be found using the same method. First, recall the definitions of the trig functions on the unit circle (Figure 1 below). The inverse sine function is defined by the property. We can re-label the diagram in terms of. The important thing to note is that. Differentiating Eq. (1) with respect to. The derivative is then,.

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Idius Land! » Tutorials & Tips

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Tutorials & Tips. What is a tensor? This section will contain tutorials and tips on mathematics, physics, Linux, programming, photography, and possibly some other topics. Basically, I'll write tips on anything that I know that other people might find helpful. I expect that this section will be the main reason most people will end up in Idius Land! After doing a web search. Thus far, I have planned 35 articles, but only 14 are finished. Is proudly powered by WordPress.