Multivariable Calculus: Introduction to functions of multiple variables

In this lecture we look at scalar-valued functions of multiple variables, e.g. z=f(x,y) and w=f(x,y,z), focusing on their notation, domain, range, and graphical representation. We explore functions mapping from ℝ^2 and ℝ^3 to ℝ, understanding how these functions translate into geometric shapes in higher dimensions. Multivariable Calculus Unit 3 Lecture 1 Key Points Real-valued functions map vectors to scalars, contrasting with the vector-valued functions we studied previously (which mapped scalars to vectors). The graph of a continuous function from ℝ^2 to ℝ is a surface in ℝ^3. Functions from ℝ^3 to ℝ cannot be graphed as the image would be in four-dimensional space. #mathematics #multivariablecalculus #calculus #surfaces #iitjammathematics #calculus3