Section 14.1 Functions of Several Variables Ruipeng Shen March 17 1 Basic Conception Definition 1. A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, {f (x, y)|(x, y) ∈ D}. Example 2. For each of the following functions, evaluate f (3, 2) then find and sketch the domain √ x+y+1 f (x, y) = x ln(y 2 − x) (a) f (x, y) = x−1 Solution (a) This function is well-defined if x+y +1 ≥ 0 and x 6= 1. The value f (3, 2) = y -1 x+y+1=0 1 -1 x x=1 √ Figure 1: The domain of f (x, y) = x+y+1 x−1 (b) This function is well-defined if y 2 > x. In addition we have f (3, 2) = 0 1 √ 6/2. y x x=y2 Figure 2: The domain of f (x, y) = x ln(y 2 − x) Example 3. Find the domain and range of g(x, y) = p 16 − x2 − y 2 . Solution This is well-defined if and only if x2 + y 2 ≤ 16. Thus the domain is a disk of radius 4. Since we know 0 ≤ 16 − x2 − y 2 ≤ 16 as (x, y) is in the domain, the range is the interval [0, 4]. y 4 x2+y2=16 4 Figure 3: The domain of f (x, y) = 2 p 16 − x2 − y 2 x 2 Graph Definition 4. If f is a function of two variables with domain D, then the graph of f is the set of the point (x, y, z) in R3 such that z = f (x, y) and (x, y) is in D. Example 5. Sketch the graph of f (x, y) = 6 − 3x − 2y. Solution This is a linear equation thus the graph is a plane. It intersects the coordinate axes at the point (2, 0, 0), (0, 3, 0) and (0, 0, 6). They are called x-intercept, y-intercept and z-intercept, respectively. z (0,0,6) z=6-3x-2y (0,3,0) (2,0,0) y x Figure 4: The graph of f (x, y) = 6 − 3x − 2y Example 6. Sketch the graph of g(x, y) = Solution p 16 − x2 − y 2 . In order to sketch the graph, we are looking for points (x, y, z) satisfying p z = 16 − x2 − y 2 =⇒ x2 + y 2 + z 2 = 16, z ≥ 0. Thus the graph is the upper half of a sphere with radius 4. Example 7. Find the domain and range and then sketch the graph of h(x, y) = 2x2 + y 2 . Solution This is well-defined for all pairs (x, y), therefore the domain is the whole plane R2 . Since x2 ≥ 0 and y 2 ≥ 0, we know h(x, y) ≥ 0 for all pairs (x, y). The range is the set of all nonnegative real numbers. The graph is an elliptic paraboloid. We can sketch it by a computer software as below. 3 Figure 5: The graph of g(x, y) = p 16 − x2 − y 2 Figure 6: The graph of h(x, y) = 2x2 + y 2 4 3 Level Curves In order to visualize more complicated two-variable functions, we can draw a contour map on which points of constant elevation are joined to form contour lines, or level curves Definition 8. The level curves of a function f of two variables are the curves with equations f (x, y) = k, where k is a constant (in the range of f ) Figure 7: The graph of the contour map of a mountain Example 9. Find approximate temperature of the locations in the figure below. 5 Figure 8: The isothermal map Example 10. Sketch the level curves of the function f (x, y) = 6 − 3x − 2y for the values k = −6, −3, 0, 3, 6, 9, 12. Solution The level set {(x, y)|f (x, y) = k} is a line 3x + 2y = 6 − k. 5 2.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 k=-6 k=-3 k=0 k=3 k=6 -2.5 k=9 k=12 -5 Figure 9: The level curves of a linear function f (x, y) = 6 − 3x − 2y 6 Example 11. Sketch the level curves of the function g(x, y) = sin(x + y). Solution For any number k ∈ [−1, 1], the level curve can be solved by sin(x + y) = k ⇐⇒ x + y = arcsin k + 2nπ or x + y = − arcsin k + (2n + 1)π Here n is any integer. In the figure below we sketch a few level curves coded by different colors. y k=1 k=0.5 k=0 k=-0.5 k=-1 x Figure 10: The level curves of the function z = sin(x + y) Figure 11: The graph of the function z = sin(x + y) 7 Example 12. Sketch some level curves of the function f (x, y) = 2x2 + y 2 . Solution The level curve is given by the equation 2x2 + y 2 = k =⇒ x2 y2 + = 1. k/2 k This is an ellipse. Thus we can sketch the graph as z=4 z z=3 z=2x2+y2 z=2 x2 y2 + =1 k/2 k y x Figure 12: The level curves and graph for the function z = 2x2 + y 2 8
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