7.2.1.3. Calculus

7.2.1.3.1. Definition of Calculus Analysis

Calculus helps you perform the deferential or integral function on the data in any curve.

Differential

Differential numerically differentiates the curve data.

Linear

The N points \(y_0=f(x_0),y-1=f(x_1),...,y_{N-1}=f(x_{N-1})\) are given. The differential of points is defined as,

\(y_i'=\frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i} (i=0,...,N-2)\)

\(y_{N-1}'=y_{N-2}'\)
Smooth

The smoothing differential evaluates the derivative by fitting a Cspline(cubic spline) to the curve data and analytically forming the derivative. Given a tabulated function \(y_i=y(x_i), i=0,...,N-1\), focus attention on particular interval, between \(x_i\) and \(x_{i+1}\). The cubic spline interpolation in that interval gives the interpolation formula \(y=Ay_i+By_{i+1}+Cy_i''+Dy_{i+1}''\).

where,

\(A \equiv \frac{x_{i+1}-x}{x_{i+1}-x_i}\)

\(B \equiv 1-A=\frac{x_-x_{i+1}}{x_{i+1}-x_i}\)

\(C \equiv \frac{1}{6}(A^3-A)(x_{i+1}-x_i)^2\)

\(D \equiv \frac{1}{6}(B^3-B)(x_{i+1}-x_i)^2\)

The first derivative of Cspline is obtained as,

\[\begin{flalign} \frac{dy}{dx}=\frac{y_{i+1}-y_i}{x_{i+1}-x_i}-\frac{3A^2-1}{6}(x_{i+1}-x_i)y_i''+\frac{3B^2-1}{6}(x_{i+1}-x_i)y_{i+1}'' \end{flalign}\]

Integral

Integral numerically integrates the curve data.

Linear

The N points \(y_0=f(x_0),y_1-f(x_1),...,y_{N-1}=f(x_{N-1})\) are given. The integral of points is defined as,

\(Y_i=\sum _{j=1}^{i}(x_i-x_{i-1}) \times y_i (i=1,...,N-1)\)

\(Y_0=0\)
Smooth

The smoothing integral evaluates the integral by fitting a Cspline(cubic spline) to the curve data and analytically forming the integral. Given a tabulated function \(y_i=y(x_i), i=0,...,N-1\), focus attention on particular interval, between \(x_i\) and \(x_{i+1}\). The cubic spline interpolation in that interval gives the interpolation formula \(y=Ay_i+By_{i+1}+cy_i''+Dy_{i+1}''\).

where,

\(A \equiv \frac{x_{i+1}-x}{x_{i+1}-x_i}\)

\(B \equiv 1-A=\frac{x_-x_{i+1}}{x_{i+1}-x_i}\)

\(C \equiv \frac{1}{6}(A^3-A)(x_{i+1}-x_i)^2\)

\(D \equiv \frac{1}{6}(B^3-B)(x_{i+1}-x_i)^2\)

The integral of Cspline is obtained as,

\(Y_i=\sum _{i=0}^{i} \int _{i}^{i+1} {ydx}=\sum _{i=0}^{i} \left[ \begin{matrix} \left[ y_i \times \frac{x_{i+1} \times x-x^2/2}{x_{i+1}-x_i} \right]_{x_i} ^{x_{i+1}} + \left[ y_{i+1} \times \frac{x^2/2-x_i \times}{x_{i+1}-x_i}\right]_{x_i} ^{x_{i+1}} \\ +\left[ \frac{y_i''}{6} \times (x_{i+1}-x_i)^2 \times \left( \frac{A^4}{4}-\frac{A^2}{2} \right) \times (x_i-x_{i+1}) \right]_{x_i} ^{x_{i+1}} \\ + \left[ \frac{y_{i+1}''}{6} \times (x_{i+1}-x_i)^2 \times \left( \frac{B^4}{4}- \frac{B^2}{2}\right) \times (x_{i+1}-x_i) \right]_{x_i} ^{x_{i+1}} \end{matrix} \right] (i=1,...,N-1)\)

\(Y_0=0\)

Differential / Integral

Calculus Type: Selects a type of Calculus.
- Differential
- Integral
Source Curve: Selects a curve.
Plot to New Page: If the user wants to draw to a new page, select Yes. If the user wants to draw to the current page, select No. (The default option is No.)
Add to Database: If the user wants to add a desired result to the database, select Yes. (The default option is No.)
Use Default Curve Name: If you want to use the default curve name like “ADD(Acc_TM-Body1(mm/s^2), Vel_TM-Body1(mm/s))”, select Yes. If not, the Chart use the Curve Name.
Curve Name: If Use Default Curve Name is No, Chart use this for a name.

Calculus Option

Option Type
Linear: The Linear curve evaluates by given points
Smooth: The smoothing integral evaluates the integral by fitting a Cubic Spline.