Multivariable Calculus

In Chap.  4 , we have studied the calculus of functions of one variable, e.g., functions expressed by \(y=f(x)\) . Basically, we studied curves in a 2D plane, the tangent to a curve at any point on the curve (first derivative) ,and the area under the curve (integral). Now is the time to the real world: functions of multiple variables. We will discuss functions of the form \(z=f(x,y)\) known as scalar-valued functions of two variables. A plot of \(z=f(x,y)\) gives a surface in a 3D space. Of course, we are going to differentiate \(z=f(x,y)\) and thus partial derivatives \( \frac{\partial {f} }{\partial {x} } \) , \( \frac{\partial {f} }{\partial {y} } \) naturally emerge. We also compute integrals of \(z=f(x,y)\) , the double integrals \(\iint f(x,y)dxdy\) which can be visualized as the volume under the surface f(x, y). And triple integrals \(\iiint f(x,y,z)dxdydz\) appear when we deal with functions of three variables f(x, y, z). All of these are merely an extension of the calculus we know from Chap.  4 . If there are some difficulties, they are just technical not mentally as when we learned about the spontaneous speed of a moving car.