dkayMath - How to Use

Once all necessary input is entered, simply hit the "Calc" button to carry out the calculation.

Currently this utility supports four types of input: matrices, multivariable equations f(x,y,z) , parametric equations f(u(x,y,z),v(x,y,z),w(x,y,z)), and differential equations.

Note: This is a beta version and not all answers can be guaranteed. Not for use in critical systems.

__Matrices__

**Input**

Select Matrix from the options menu and enter in each entry of the matrix followed by a comma, excluding the last entry. Each entry can be a multivariable equation. The order to enter in the matrix will be to the first row, then the second, then the third... The program will automatically (try to) assume that the matrix is square when no row/column input is entered in the row/column text fields. If the input matrix is however not square, enter in the number of rows in the first text box to the right of Rows/Columns, followed by the number of columns in the second text box. For example:

1 2 3

3 7 3

0 1 1 = 1,2,3,3,7,3,0,1,1 as the input, and Rows/Columns can either be left blank or set to 3 and 3.

1 2 3

1 1 0 = 1,2,3,1,1,0 as the input, and Rows/Columns set to 2 and 3.

2-t 3 1

1 1-t 0

3 9 -2-t = 2-t,3,1,1,1-t,0,3,9,-2-t as the input, and Rows/Columns can be left blank.

**Current Actions:**

- echelonReduce the given matrix into echelon form.

- redEchReduce the given matrix into reduced echelon form.

- inverseWill find inverse of given matrix based on reducing A|I. Will only attempt if determinant of A is nonzero (required for invertibility). Requires a square matrix.

Note: Could see loss of precision in large matrices. Also if variables are used in the inputted matrix and answer is not achieved or seems off, try using the invAdj action to find the inverse instead.

- invAdjFind the inverse of the given matrix by using A
^{-1}= 1/det(A) Adj(A). Requires a square matrix.

Note: This method will usually work better with matrices involving variables.

- adjFind the adjoint matrix, where Adj(A) is the transpose of the matrix of cofactors. Requires a square matrix.

- charPolyFind the characteristic polynomial of the given matrix, which is the determinant of A-tI.

- eigenCurrently will find all eigenvalues, eigenvectors, and eigenspaces for a 2x2 matrix. Will find all real eigenvalues, eigenvectors, and eigenspaces for a 3x3 matrix.

- detFinds the determinant of given matrix. Works up to 5x5. Must be square matrix.

- nullspaceFinds nullspace of given matrix.

- spbasisFinds basis for spanning set of given matrix.

Note: A vector space does not a have unique basis, so answers may vary. However, each basis will have the same number of vectors.

- rsbasisFinds basis for the row space of given matrix.

- rangeFinds range for the given matrix.

- LUFinds the LU factorization of A where A = LU, L being a lower triangular matrix, and U being upper triangular.

- hessFinds Hessenberg form H of a matrix A where H is similar to A, i.e. H = SAS
^{-1}.

- hessPolyFinds characteristic polynomial of the unreduced Hessenberg matrix A.

- houseFinds Householder transform Q of the Matrix A, where Q is a symmetric matrix defined by Q = I - (2/u
^{T}u)*uu^{T}.

- QRFinds QR factorization of given matrix, Q orthogonal, and R upper triangular.

- sqauresFinds least-squares solution x* to the equation Ax=b, where x* is found by solving A
^{T}Ax* = A^{T}b. Original matrix must be entered as A|b.

__Multivariable equations f(x,y,z)__

**Input**

Select Fx from the options menu and enter in an equation much the same way as you would on a graphing calculator, for example:

3xy

^{2}= 3xy^2 , 7z-3z

^{(3x+4)}= 7z-3z^(3x+4) , x

^{3}ycos(z) = x^3ycos(z) , y

^{2}ln(z) = y^2ln(z) ...

When entering an equation with numerators and denominators, separate each with parenthesis, for example:

(x)/(y) , (ycos(z))/(x+7) , (y^(3z))/(e(x)) ....

**Specific Formats:**

Cosine : cos(x) , cos(x)cos(x) , xycos(z)....

Sine : sin(y) , sin(x^2) , sin(xy+3)....

Natural Log : ln(x) , xln(x) , x^2ln(y+z) ....

Exponential : e(x) , e(xy+4) , xe(yz)cos(y)....

π : cos(PI) , sin(3.5PI) , e(x)sin(-3PI)....

Secant : sec(x) , sec(y^2) , sec(x)cos(y)....

Cosecant : csc(x) , xcsc(x) , csc(x)ln(y)....

Tangent : tan(x) , t^2tan(t) , tan(xz)ln(x)....

Cotangent : cot(x) , cot(y)cos(x) , z^2cot(z)....

**Current Actions:**

- diffDifferentiate the function based on variable chosen from Var dropdown menu. Will handle common differentiation, product rule, quotient rule.

- integrateIntegrate the function based on the variable chosen from Var dropdown menu. Fill in the integration values to integrate over two values. The first box will the value to be integrated from, the second being the value to integrate to. If the boxes are left blank, indefinite integration will be carried out. Will handle common integration,integration by parts.

- multiplyMultiply two functions together. Separate each function by the asterisk symbol, *. For example: x^2*3xy.

- rootsFinds the zeros of the given quadratic or cubic polynomial. For quadratic polynomials it will find both real and non-real roots. For cubic polynomials it will only find the real roots currently.

__Parametric equations f(u(x,y,z),v(x,y,z),w(x,y,z)) or f = P__

**i**+ Q**j**+ R**k****Input**

Select Parametric from the options menu and enter in a parametric equation by separating each function by a comma. If the z function in w(x,y,z) is left blank, a 0 will be assumed. For example:

3xy^2,yz,z cos(t),sin(t),t xln(y),z,y sin(t),cos(t) ....

**Current Actions:**

- gradientReturns the gradient of given parametric equation f(u,v,w), where ∇f = ( ∂u/∂x , ∂v/∂y, ∂w/∂z ).

- isGradReturns the function F(x,y,z) such that the gradient of F is f, or ∇F = f. Will only work if the parametric equation entered f = P
**i**+ Q**j**+ R**k**is conservative, meaning ∂P/∂y = ∂Q/∂x.

- curlReturns the curl of our function f = P
**i**+ Q**j**+ R**k**, where curl(f) = (∂R/∂y - ∂Q/∂z)**i**+ (∂P/∂z - ∂R/∂x)**j**+ (∂Q/∂x - ∂P/∂y)**k**.

- divReturns the divergence of our function f = P
**i**+ Q**j**+ R**k**, where div(f) = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

- magReturns the magnitude of the given vector/parametrization A, |A| = (a
_{1}^{2}+ a_{2}^{2}+ a_{3}^{2})^{1/2}.

__Differential Equations__

**Input**

Select Diff Eq from the options menu and enter in a differential equation. The input is assumed to have the form of ay''+by'+c -G(x) = 0, so no equal sign is required. For example:

y''+2y'+3 , 3y''-y , -3y''-y-x^2 , 3y''-y'+5y-xe(x)-cos(x) ...

**Current Actions:**

- solveSOSolves homogenous and nonhomogenous second order differential equations for y.

- IVP(SO)Solves second order initial value problems. Separate the differential equation from the initial values by commas.