Eigenvalues and Eigenvectors of Matrices

Mathematical Formulation Consider a transformation matrix $latex {A}&s=1$, such that it transforms a vector $latex {X}&s=1$ into $latex {Y}&s=1$. Thus, we can write, $latex {Y = AX}&s=2$ Recall that 2 vectors directed along same direction are simply scalar multiples of each other. e.g. $latex {\vec P = 3 \hat i + 4 \hat j =… Continue reading Eigenvalues and Eigenvectors of Matrices

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Errors and Approximations, Maxima and Minima, Lagrange’s Method

Errors and Approximations If $latex {f}&s=1$ is a function of $latex {x,y,z}&s=1$, then the error in $latex {f}&s=1$ is $latex {df = \frac {\partial f}{\partial x} dx + \frac {\partial f}{\partial y} dy + \frac {\partial f}{\partial z} dz}&s=2$ However, this is an approximation. Note: If necessary, we take log on both sides. Maxima and… Continue reading Errors and Approximations, Maxima and Minima, Lagrange’s Method

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Jacobians

Introduction To get an idea about Jacobians, one requires knowledge of matrices, determinants, functions and partial differentiation. I) Matrices and Determinants : A matrix is a rectangular arrangement of numbers in $latex {m}&s=1$ rows and $latex {n}&s=1$ columns. When $latex {m =n}&s=1$, the matrix is known as a square matrix. Determinants are defined for square… Continue reading Jacobians

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Partial Differential Equations

Introduction So far, while studying calculus, we have dealt with functions of single variable, i.e. $latex {y=f(x)}&s=1$. $latex {sin (x^2), ln \ x, e^{cos \ (tan \ x)}}&s=1$ are few examples. Irrespective of their complexity, the variable $latex {y}&s=1$ always depended on the value of independent variable $latex {x}&s=1$. We also defined the derivatives and… Continue reading Partial Differential Equations