Okay this may sound stupid but I need a little help... What do $\Large \frac{d}{dx}$ and $\Large \frac{dy}{dx}$ mean? I need a thorough explanation. Thanks.

General Solution: A solution of a differential equation in which the number of arbitrary constants is equal to the order of the differential equation is called the general solution or complete integral or complete primitive. Singular solution: A solution which can not be obtained f...

What is the geometric significance of substitution? For instance substituting $x = r\cos(\theta)$ and $y = r\sin(\theta)$ in the following problem to find the limit. $\lim\limits_{(x,y) \to (0,0)} \dfrac{\tan(x^2 + y^2)}{x^2+y^2}$. Of course, this manipulation/substitution makes it easier to sol...

Suppose you have a 2x2 real-valued matrix, $\mathbf{A}$. If you perform a singular value decomposition (SVD), then this can be understood geometrically as a decomposition of $\mathbf{A}$ into a 2-D rotation, scaling and second 2-D rotation of the form: $$\mathbf{A} = \mathbf{R_1 S R_2}$$ Howeve...

In the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum. But what about the g...