Function of several variables”, “Multivariate function”, and “Multivariable function” schaum’s advanced calculus pdf here. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified. This article will use bold.

Moreover, many functions are defined in such a way that it is difficult to specify explicitly their domain. One can easily obtain a function in one real variable by giving a constant value to all but one of the variables. In next section, we will show that, if the multivariable function is continuous, so are all these univariable functions, but the converse is not necessarily true. As continuous functions of several real variables are ubiquitous in mathematics, it is worth to define this notion without reference to the general notion of continuous maps between topological space. A function is continuous if it is continuous at every point of its domain. If the limit exists, it is unique. A first derivative is positive if the function increases along the direction of the relevant axis, negative if it decreases, and zero if there is no increase or decrease.

Evaluating a partial derivative at a particular point in the domain gives the rate of change of the function at that point in the direction parallel to a particular axis, a real number. Partial derivatives extend this idea to tangent hyperplanes to a curve. While bounded hypervolume is a useful insight, the more important idea of definite integrals is that they represent total quantities within space. 2D surface in 3D Euclidean space. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.