Math is quite abstract but it has immense usage in our life. Multivariable calculus essentially helps solve complicated problems in practice such as simple needs to measure an irregular area, volume by approximating.

To do a good job of approximating, first, introduce the concept of local linearilization by first dirivative for simple x, y function.

Lf(x0,y0) = a f(x0, y0) partial derivative with respect to x (x-x0) + bf(x0 ,y0) partial derivative with respect to y(y-y0) + constant(f(x0,y0). This complex algebra form can be compactly expressed in vector language below:

Analogously, this simple linear form is inadequate to treat convoluted surface/volume, so quadratic form is necessary, where higher level of derivatives are included.

Again, we try to express it in a succinct way that tens of hundreds of potential variables can be exhibited with the help of Hessian matrix and matrix multiplication rule: