Curves in R2: Graphs vs Level Sets Graphs (y = f(x)): The graph of f : R ! R is 2 f(x; y) 2 R j y = f(x)g: Example: When we say \the curve y = x2," we really mean: \The graph of the function f(x) = x2." That is, we mean the set f(x; y) 2 R2 j y = x2g. Level Sets (F (x; y) = c): The level set of F : R2 ! R at height c is 2 f(x; y) 2 R j F (x; y) = cg: Example: When we say \the curve x2 + y2 = 1," we really mean: \The level set of the function F (x; y) = x2 + y2 at height 1." That is, we mean the set f(x; y) 2 R2 j x2 + y2 = 1g. Note: Every graph is a level set (why?). But not every level set is a graph. Graphs must pass the vertical line test. (Level sets may or may not.) Surfaces in R3: Graphs vs Level Sets Graphs (z = f(x; y)): The graph of f : R2 ! R is 3 f(x; y; z) 2 R j z = f(x; y)g: Example: When we say \the surface z = x2 + y2," we really mean: \The graph of the func- tion f(x; y) = x2 + y2." That is, we mean the set f(x; y; z) 2 R3 j z = x2 + y2g. Level Sets (F (x; y; z) = c): The level set of F : R3 ! R at height c is 3 f(x; y; z) 2 R j F (x; y; z) = cg: Example: When we say \the surface x2 + y2 + z2 = 1," we really mean: \The level set of the function F (x; y; z) = x2 + y2 + z2 at height 1." That is, f(x; y; z) 2 R3 j x2 + y2 + z2 = 1g. Again: Every graph is a level set (why?). But not every level set is a graph. Graphs must pass the vertical line test. (Level sets may or may not.) Q: Do you see the patterns here? For example, suppose G: R7 ! R. ◦ What does the graph of G look like? (A: 7-dimensional object in R8. 8 That is, f(x1; : : : ; x7; x8) 2 R j x8 = G(x1; : : : ; x7)g.) ◦ What do the level sets of G look like? (A: They are (generically) 6-dim 7 7 objects in R . That is, f(x1; : : : ; x7) 2 R j G(x1; : : : ; x7) = cg.) Curves in R2: Examples of Graphs f(x; y) 2 R2 y = ax + bg: Line (not vertical) f(x; y) 2 R2 y = ax2 + bx + cg: Parabola Curves in R2: Examples of Level Sets f(x; y) 2 R2 ax + by = cg: Line x2 y2 (x; y) 2 2 + = 1 : Ellipse R a2 b2 x2 y2 (x; y) 2 2 − = 1 : Hyperbola R a2 b2 x2 y2 (x; y) 2 2 − + = 1 : Hyperbola R a2 b2 Surfaces in R3: Examples of Graphs n o 3 (x; y; z) 2 R z = ax + by : Plane (not containing a vertical line) x2 y2 (x; y; z) 2 3 z = + : Elliptic Paraboloid R a2 b2 x2 y2 (x; y; z) 2 3 z = − − : Elliptic Paraboloid R a2 b2 x2 y2 (x; y; z) 2 3 z = − : Hyperbolic Paraboloid (\saddle") R a2 b2 x2 y2 (x; y; z) 2 3 z = − + : Hyperbolic Paraboloid (\saddle") R a2 b2 Surfaces in R3: Examples of Level Sets n o 3 (x; y; z) 2 R ax + by + cz = d : Plane x2 y2 z2 (x; y; z) 2 3 + + = 1 : Ellipsoid R a2 b2 c2 x2 y2 z2 (x; y; z) 2 3 + − = 1 : Hyperboloid of 1 Sheet R a2 b2 c2 x2 y2 z2 (x; y; z) 2 3 − − = 1 : Hyperboloid of 2 Sheets R a2 b2 c2 Two Model Examples Example 1A (Elliptic Paraboloid): Consider f : R2 ! R given by f(x; y) = x2 + y2: The level sets of f are curves in R2. Level sets are f(x; y) 2 R2 x2 +y2 = cg. The graph of f is a surface in R3. Graph is f(x; y; z) 2 R3 z = x2 + y2g. Notice that (0; 0; 0) is a local minimum of f. @f @f @2f @2f Note that @x (0; 0) = @y (0; 0) = 0. Also, @x2 (0; 0) > 0 and @y2 (0; 0) > 0. Sketch the level sets of f and the graph of f: Example 1B (Elliptic Paraboloid): Consider f : R2 ! R given by f(x; y) = −x2 − y2: The level sets of f are curves in R2. Level sets are f(x; y) 2 R2 −x2−y2 = cg. The graph of f is a surface in R3. Graph is f(x; y; z) 2 R3 z = −x2 − y2g: Notice that (0; 0; 0) is a local maximum of f. @f @f @2f @2f Note that @x (0; 0) = @y (0; 0) = 0. Also, @x2 (0; 0) < 0 and @y2 (0; 0) < 0. Sketch the level sets of f and the graph of f: Example 2 (Hyperbolic Paraboloid): Consider f : R2 ! R given by f(x; y) = x2 − y2: The level sets of f are curves in R2. Level sets are f(x; y) 2 R2 x2 −y2 = cg. The graph of f is a surface in R3. Graph is f(x; y; z) 2 R3 z = x2 − y2g: Notice that (0; 0; 0) is a saddle point of the graph of f. @f @f @2f @2f Note that @x (0; 0) = @y (0; 0) = 0. Also, @x2 (0; 0) > 0 while @y2 (0; 0) < 0. Sketch the level sets of f and the graph of f: Introduction: Derivatives of Functions Rn ! Rm Def: Let F : Rn ! Rm be a function, say F (x1; : : : ; xn) = (F1(x1; : : : ; xn);:::;Fm(x1; : : : ; xn)): Its derivative at the point (x1; : : : ; xn) is the linear transformation n m DF (x1; : : : ; xn): R ! R whose (m × n) matrix is 2 3 @F1 ··· @F1 @x1 @xn 6 . 7 DF (x1; : : : ; xn) = 4 . 5: @Fm ··· @Fm @x1 @xn Note: The columns are the partial derivatives with respect to x1, then to 1 2 x2, etc. The rows are the gradients of the component functions F , F , etc. 2 3 2 3 rF1 DF (x ; : : : ; x ) = @F ··· @F = . : 1 n 4@x1 @xn 5 4 . 5 rFm Example: Let F : R2 ! R3 be the function y F (x; y) = (x + 2y; sin(x); e ) = (F1(x; y);F2(x; y);F3(x; y)): Its derivative at (x; y) is a linear transformation DF (x; y): R2 ! R3. The matrix of the linear transformation DF (x; y) is: 2@F1 @F1 3 2 3 @x @y 1 2 DF (x; y) = 6@F2 @F2 7 = cos(x) 0 : 4 @x @y 5 4 5 @F3 @F3 0 ey @x @y Notice that (for instance) DF (1; 1) is a linear transformation, as is DF (2; 3), etc. That is, each DF (x; y) is a linear transformation R2 ! R3. Goals: We will: ◦ Interpret the derivative of F as the \best linear approximation to F ." ◦ State a Chain Rule for multivariable derivatives. Introduction: Gradient of Functions Rn ! R Def: Let f : Rn ! R be a function. n Recall: The derivative of f : R ! R at the point x = (x1; : : : ; xn) is the 1 × n matrix h i Df(x) = @f ··· @f : @x1 @xn n The gradient of f : R ! R at the point x = (x1; : : : ; xn) is the vector h i rf(x) = @f ;:::; @f : @x1 @xn n The directional derivative of f : R ! R at the point x = (x1; : : : ; xn) in the direction v 2 Rn is the dot product of the vectors rf(x) and v: Dvf(x) = rf(x) · v: We will give geometric interpretations of these concepts later in the course. Introduction: Hessian of Functions Rn ! R Theorem: Let f : Rn ! R be a function whose second partial derivatives are all continuous. Then: @2f @2f = : @xi@xj @xj@xi In brief: \Second partials commute." Def: Let f : Rn ! R be a function. n The Hessian of f : R ! R at the point x = (x1; : : : ; xn) is the n × n matrix 2 @2f @2f 3 2 ··· @x1 @x1@xn 6 . 7 Hf(x) = 4 . 5: @2f @2f ··· 2 @xn@x1 @xn The directional second derivative of f : Rn ! R at the point x = n (x1; : : : ; xn) in the direction v 2 R is vT Hf(x) v: Again, we will give geometric interpretations of these concepts later on. Linear Approximation: Single-Variable Calculus Review: In single-variable calc, we look at functions f : R ! R. We write y = f(x), and at a point (a; f(a)) write: ∆y ≈ dy: Here, ∆y = f(x) − f(a), while dy = f 0(a)∆x = f 0(a)(x − a). So: f(x) − f(a) ≈ f 0(a)(x − a): Therefore: f(x) ≈ f(a) + f 0(a)(x − a): The right-hand side f(a) + f 0(a)(x − a) can be interpreted as follows: ◦ It is the best linear approximation to f(x) at x = a. ◦ It is the 1st Taylor polynomial to f(x) at x = a. ◦ The line y = f(a) + f 0(a)(x − a) is the tangent line at (a; f(a)). Linear Approximation: Multivariable Calculus Now consider functions f : Rn ! Rm. At a point (a; f(a)), we have exactly the same thing: f(x) − f(a) ≈ Df(a)(x − a): That is: f(x) ≈ f(a) + Df(a)(x − a): (∗) Note: The object Df(a) is a matrix, while (x − a) is a vector. That is, Df(a)(x − a) is matrix-vector multiplication. 2 Example: Let f : R ! R.