The graph of a function of two variables is a surface in ℝ 3 and can be studied using level curves and vertical traces A set of level curves is called a contour map

To study limits and continuity for functions of two variables, we use a δ disk centered around a given point A function of several variables has a limit if for any point in a δ ball centered at a point P , the value of the function at that point is arbitrarily close to a fixed value (the limit value)

partial derivative — a derivative of a function of more than one independent variable in which all the variables but one are held constant. vertical trace — the set of ordered triples ( c , y , z ) that solves the equation f ( c , y ) = z for a given constant x = c or the set of ordered triples ( x , d , z ) that solves the equation…. function of two variables — a function z = f ( x , y ) that maps each ordered pair ( x , y ) in a subset D of ℝ 2 to a unique real number z. δ ball — all points in ℝ 3 lying at a distance of less than δ from ( x 0 , y 0 , z 0 ).