Math -- 3DSoftware.com

Math

General Reference

The following concepts are used in our documentation.

Unless otherwise specified, right handed rectangular coordinates with orthogonal (perpendicular) coordinate axes are assumed.

In two dimensions, the first axis points to the right (East) and the second axis points up (North). The example at right shows X as the first coordinate axis and Y as the second coordinate axis.

For three dimensions, a third dimension points straight out from the plane of the first two dimensions, corresponding to elevation height above that plane. Using flat ground as the plane of the first two dimensions, with the first dimension pointing East and the second pointing North, then a positive value for the third dimension is above ground, and negative for below ground.

Intervals

Intervals (in R¹) are denoted with parenthesis marking an endpoint that is not included in the interval, and square bracket marking an end point that is included in the interval. All possible combinations are:

[a, b]
(a, b)
[a, b)
(a, b]

:
:
:
:

a ≤ x ≤ b
a < x < b
a ≤ x < b
a < x ≤ b

Slope of a Line

The slope of a line is the ratio by which the line is rising or falling as it is plotted from left to right (in the positive X direction).

The slope is the ratio of rise over run (goings), which is rise divided by run: rise / run

Given two points P and Q, with Q to the right of P, the slope m of the line that passes through P and Q is:

If the line is horizontal, the slope is zero. If the line is vertical, the slope is infinite (undefined). The slope is positive if the line moves upward, or negative if it points downward:

Those lines are perpendicular. For perpendicular lines, the slopes are negative reciprocals of each other:

Slope on a Curve

In mathematics, a straight line is a type of curve, but not all curves are straight lines.

The slope of a point on a curve is known as the tangent of the curve at that point. The tangent is a straight line. If the curve is a straight line, it coincides with the tangent.

For some curves you can use geometry, algebra and/or trigonometry to find the tangent of a point. For example, to find the tangent of a point A on a circle, find two relatively nearby points B and C on the circle, each equal distance away from point A (on opposite sides of A). Draw a straight line through B and C, then find the straight line passing through A that is parallel to the BC line.

But that is cumbersome, and cannot be done for many types of curves, even continuous curves. A better method is to use differential calculus to find the derivative function of the curve, and use that to calculate the slope or tangent for any point on the curve. This is possible for many types of curves.

The Derivative

The derivative of a point on a curve is the slope (or tangent) of the curve at that point. This slope represents the rate of change of the curve at that point.

As an example, we will find the slope of a point P = ( 1, 1 ) on a parabola that has this equation:

That parabola opens upward. Assume we have a point Q that is also on the parabola, above and to the right of P, sliding along the parabola (downwards and to the left) toward P. The distance between P and Q gets smaller as Q approaches P.

If a straight line is drawn between P and Q, that line will not be parallel to the parabola's tangent at P. But the angle between that line and P's tangent gets smaller (approaching zero) as Q approaches P.

The straight line from P to Q is called a secant. As Q approaches P, the secant's limit is P's tangent, with the secant's slope approaching the slope of the tangent. To find the slope of the tangent, we find the limit of the slope of the secant. This is expressed as follows:

The equation of the curve is y = x², thus the following substitutions can be made in the numerator above:

Py = Px² and Qy = Qx².

The numerator becomes: Qy – Py = Qx² – Px²

Factoring the numerator we get: Qx² – Px² = ( Qx – Px ) ( Qx + Px )

One of the factors of the numerator ( Qx – Px ) equals the denominator, thereby canceling out:

The limit becomes:

Since Qx becomes Px at the limit, then m = Px + Px at the limit, and Px + Px = 2Px (which is twice the value of X of a point on the parabola).

That is the derivative. The derivative of y = x² is 2x. For any point ( x, y ) on y = x², the slope of that point is 2x.

Therefore the slope of ( 1, 1 ) is 2. That is the slope of the tangent of y = x² at the point ( 1, 1 ).

This derivative, which we have visualized as the slope of a tangent, is the rate of change of the function, and is a function itself. It specifies the velocity of the function, as we progress along X. In this case, the function is x², and the derivative of that function is 2x.

Technical Note: Taking a derivative of the derivative yields the acceleration: the rate of change, of the rate of change.

A common symbol to signify the derivative is the Leibniz notation:

Each ‘d’ means “differential.” This is a single symbol, not a fraction that can be usually broken down the way actual fractions can be used. It just looks like a fraction to remind us how derivatives are derived the long way, as we did above (without shortcuts that we will explain shortly).

In other words, dy / dx is a symbol for this limit:

There are many ways to symbolize the derivative. Some of the ways to denote a derivative for a function y = f(x) are shown here:

The apostrophe is the prime mark (the last example reads “Y prime”). Higher order derivatives (derivatives of derivatives) are symbolized with a superscript or additional prime marks. The superscript is symbolic (not an exponent).

Here are two ways to denote the second derivative:

There are many formulas (rules), which can be used alone or in combination, to differentiate (find the derivative of) a function.

Here is an important rule (the power rule):

We could have used that to find the derivative of the parabola earlier in this section. Not only would this rule make that easy, but it works for any positive integer exponent.

For this function:		The derivative is:

Here is another rule for differentiating:

Combining the two rules, we calculate the second derivative:

Here is a handy rule for differentiating polynomials:

For example:

Note: A good refresher course book on this topic is Keith E. Hirst, Calculus of One Variable. See also OCW Courses.

Partial Derivatives

The partial derivative of a multivariable function is the derivative of the function with respect to one of its variables.

The function can have many variables, not just x and y, but the partial derivative is the derivative of only one of its variables. All other variables are kept constant.

“the partial derivative with respect to [a variable] is obtained by the usual rules of differentiation with the proviso that all other variables are assumed to be constant.”

—	S. Hassani, Mathematical Methods for Students of Physics and Related Fields 2nd ed., p. 48

For example, consider this function:

The partial derivative of f(x,y) with respect to x is:

The partial derivative of f(x,y) with respect to y is:

To further illustrate, we can use the letter ‘h’ to represent a constant. Using the same function, taking the partial derivative with respect to x, we make y constant. In this case, substitute ‘h’ for ‘y.’ The equation becomes:

Differentiating with respect to x:

The derivative of a constant function (in this case h²) is zero.

Substituting ‘y’ back in for ‘h’:

The Determinant

The determinant is a single number (a scalar) that provides information about a square matrix.

In linear algebra, a scalar is a number that is not a vector or matrix. Any single number is a scalar. The determinant of a matrix is a scalar.

A square matrix is a matrix that has the same number of rows and columns. Only a square matrix can have a determinant.

The determinant of a matrix A is denoted as: det(A) or |A|

The determinant is calculated by systematically combining elements of the matrix arithmetically. The precise formulas are found in math and engineering books. We provide formulas for 2 x 2 and 3 x 3 matrices in the next section (see Calculating a Determinant).

A square matrix is invertible (can have an inverse matrix, which is unique) if and only if its determinant is nonzero.

Two determinants multiplied together equal the determinant of the two matrices multiplied together: det(A)det(B) = det(AB)

The following determinant vanishes to zero when the four x,y,z points become coplanar:

Determinants can be used in computer graphics to calculate the area of certain types of transformations.

Calculating a Determinant

The determinant of a 2 x 2 matrix is:

The determinant of a 3 x 3 matrix is:

Trigonometry

Abstract Spaces

Abstractions can be levels of detail. For example, in electrical engineering:

“The main point of digital logic…is abstraction — we want to hide the underlying details and use high-level abstractions whenever possible.”

—	D.E. Comer, Essentials of Computer Architecture, p. 29

In mathematics, abstractions form spaces, like namespaces and classes in C++:

“A namespace is a logical unit that contains related declarations and definitions.”

—	D. Yang, C++ and Object-Oriented Numeric Computing, p. 113

“The only requirement in a definition is that it be consistent, i.e. that the various conditions do not contradict one another or the rules of mathematics. Apart from this there is freedom in the choice of conditions”

—	S. Dineen, Multivariate Calculus and Geometry 2nd ed., p. 43

Saturday, 31-Jul-2010 16:11:45 GMT