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Three-dimensional Analytic Geometry and Vectors

Chapter 11. Three-dimensional Analytic Geometry and Vectors. up down return end. 11.5 Partial derivatives and total differential.

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Three-dimensional Analytic Geometry and Vectors

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  1. Chapter 11 Three-dimensional Analytic Geometry and Vectors up down return end

  2. 11.5Partial derivatives and total differential 1.Let f (x, y) be a function of two variables, and suppose that x vary while keeping y fixed, say y=b, where b is a constant. Then we have a function of single variable x, namely g(x)= f (x, b). If g(x) has a derivative at a, then we call the derivative the partial derivative of f with respect to x at (a, b) and denoted it by fx(a, b). Thus we have fx(a,b)=g'(a) where g(x)= f (x, b) or up down return end

  3. similarly, is called partial derivative of f(x, y)with respect y at(a, b). 2. If let the point (a,b) vary, fxand fy become functions of two variables. So we can get the following: If fxis a function of two variables, its partial derivatives are functions fxand fy defined by fx(x,y)= fy(x,y)= up down return end

  4. There are other notations for partial derivatives:If z=f(x, y), fx(x,y)= fx= f1=D1f =Dxf = fy(x,y)=fy=f2 = D2f =Dyf = Rule for finding partial derivatives ofz=f(x,y): <1>. To find fx , regard y as a constant and differentiate f(x, y) with respect with x. <2>. To find fy , regard x as a constant and differentiate f(x, y) with respect with y. up down return end

  5. ExampleIf f(x, y)=xsin(x2+y2)exy, find fx(2, 1) and fy(1, 2). ExampleIf f(x, y)=cos( ), find fx(x, y) and fy(x, y). Similarly, we can define partial derivatives of function of more than two variables. If u=f(x1, x2 , ..., xn), then i=1,2,....,n up down return end

  6. 3.We can also define fxx=(fx )x , fxy=(fx )y , and so on. fxx= fxy= up down return end

  7. DefinitionIf z=f(x, y), then f is called differential at (a, b) if z= fx(a, b) x+ fy(a, b) y+1x+2y where 1 and 20 as (x,y)0 . Then generally fx(x, y)x+ fy(x, y)y is called total differential, denoted by dz or df . That is dz=df = fx(x, y)x+ fy(x, y)y ExampleIf z=f(x, y)=2x2+y2, find total differential dz. Similarly, we can define total differential of function of more than two variables. up down return end

  8. 11.6The Chain Rule 1. The Chain rule (case I) Suppose that z=f(x, y) is a differentiable function of x and y, where x=g(t) and y=h(t) are both differentiable functions of t. Then z is a differentiable function of t and Proof A change of t produces changes of x in x and y in y. Then it produces change of z in z,and from the definition we have up down return end

  9. We let t0, then x=g(t+t) - g(t) 0，because g is differentiable and therefore continuous. Similarly, y0. This means that 10 , 20 , so 2. If z=x2y+3xy4, where x=et , y=sint , find dz/dx. 3.Suppose chain rule (case 2) Suppose that z=f(x, y) is differentiable, x=g(t, s), y=h(t, s), and the partial derivatives, gt,gs ,ht and hs exist. Then up down return end

  10. 4. The chain rule (General version). Suppose u=f(x1,x2 ,... ,xn), and each xi is a function of m variables t1, t2 ,... , tm , such that the all partial derivatives, xj ti,exist (j=1, 2,...., m). Then u is a function of t1, t2 ,... , tm , and for , j=1,2,.... ,m. 5. If u=x4y+y2z3, where x=rset,y=rs2e-t , z=r2ssint , find u s when r=2, s=1, t=0. up down return end

  11. 6. Implicit differentiation Wesuppose that an equation of the form F(x, y)=0, define y implicitly as a differentiable function of x , that is , y=f(x), where F(x, f(x))0 for all x in the domain of f. If is F(x, y) is differentiable and F y0, then 7. Example Find y ' if x3+y3=6xy. up down return end

  12. 9. Example Find and if x3+y3 +z3 =6xy. 8. More generally, Wesuppose that z is given implicitly by an equation of the form F(x, y, z)=0. This means that F(x, y, f(x, y))0 for all (x, y) in the domain of f. If F(x, y, z) is differentiable and F z0, then up down return end

  13. 11.7Directional derivatives and the gradient Definition The directional derivative of at (x0, y0) in the direction of a unit vector u=<a, b> is if this limit exists. Theorem If f is differentiable function of x and y, then f has directional derivative in the direction of any vector unit vector u=<a, b> and up down return end

  14. Example Find the directional derivative Duf(x, y) if f(x, y)=x3-3xy+4y2 and u is the unit vector given by angle =/6. What is Duf(1, 2) ? Definition If f is a function of two variables x and y, then the gradient of f is the vector function f(x, y) defined by f(x, y)=< fx(x, y) , fy(x, y) >= up down return end

  15. Example If f(x, y)=sinx+exy , find the f(x, y). Corollary u=<a,b> , Duf(x, y) =f(x, y)·u. Example If f(x, y)=x2 y3–4y , find the directional derivative of f(x, y) at (2, –1) in the direction v=2i+5j,. up down return end

  16. For functions of three variables we can define directional derivatives in a similar manner. Definition The directional derivative of at (x0, y0 , z0) in the direction of a unit vector u=<a, b, c> is if this limit exists. We can use vector notation as following: P0=< x0, y0 , z0 > (or P0=< x0, y0> ) and up down return end

  17. Definition If f is a function of three variables x, y and z, then the gradient of f is the vector function, denoted by f(x, y) or gradf, defined by gradf =f(x, y)=< fx , fy, fz >= Corollary: u=<a, b, c> , Duf(x, y, z) =f(x, y, z)·u. Example If f(x, y, z)=xsin(zy), find (a) the directional derivative of f(x, y, z) at (1, 3, 0) in the direction v=1i+2j–k. (b) the gradient of f . up down return end

  18. Theorem Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative Duf(P) is |f(P)| and it occurs when u has the same direction as the gradient vector f(P) . Example (a) If f(x, y)=xey, find the rate of change at p(2, 0) in direction from P to Q(1/2, 2). (b)In what direction does f have the maximum rate of change ? What is this maximum rate of change. p807 up down return end

  19. Tangent planes to level surface: (ommited)

  20. 11.8Maximum and minimum values p812 Definition A function of two or three variables has a local maximum at (a, b) (or (a, b, c)) if f(x, y) f(a, b) (or f(x, y, z) f(a, b, c) ) for all (x, y) (or (x, y, z)) in such a small disk with center (a, b) (or (a, b, c)) . If f(x, y) f(a, b) (or f(x, y, z)  f(a, b, c) ) for all (x, y) (or (x, y, z)) in such a small disk with center (a, b) (or (a, b, c)) , f(a, b) (or f(a, b, c) ) is a local minimum value. If for all points (x, y) (or (x, y, z)) in the domain of f , then has an absolute maximum (or absolute minimum) at point (a, b) (or (a, b, c)) . up down return end

  21. Theorem If f has a local extremum (this is local minimum or local maximum) (a,b) and the first-order partial derivatives of f exists there, then fx(a,b)=0 and fy(a,b)=0 . A point (a,b) such that fx(a,b)=0 and fy(a,b)=0, or one of these partial derivatives does not exist, is called critical point (or stationary point) . up down return end

  22. Example Find the extremum values of f(x,y)=y2-x2, f(0,0)=0 can not be an extremum value for f, so f has no extremum values. But (0,0) is a the critical point of f. Example Find the extremum values of f(x,y)=y2+x2 -2x-6y+14. up down return end

  23. The Second derivative test Suppose the second partial derivatives of f are continuous in a disk with center (a, b) and (a, b) is the critical point. Let D=D(a,b) = fxx(a, b) fyy(a, b) –[fxy(a, b)]2 (a) If D>0 and fxx(a, b)>0, then f(a, b) is a local minimum. (b) If D>0 and fxx(a, b)<0, then f(a, b) is a local maximum. (c) If D<0, then f(a, b) is not extremum, then (a, b) is called a saddle point. up down return end

  24. Example Find the local extremum of f(x,y)=x4+y4 -4xy+1. up down return end

  25. 11.9Lagrange multiplier P821 Suppose now that we want to find the maximum and minimum values of f(x, y, z) subject to two constrain (side conditions) of form g(x, y, z)=k and h(x, y, z)=c. Example Find the maximum value of the function V=xyz, subject to the constrain 2xz+2yz+xy=12. up down return end

  26. To find the the maximum and minimum values of f(x,y,z) subject to two constrain g(x,y,z)=k and h(x,y,z)=c(assuming that the extrema exist): • Find all values of x, y, z, and , , such that f(x, y, z)=  g(x, y, z)+  h(x, y, z) and g(x, y, z)=k, h(x, y, z)=c; • Evaluate f at all the points (x, y, z) that arise from step • (a). The largest of these values is the maximum value of f . The smallest of these values is the minimum value of f . ( and  are called Lagrange multiplier. This method is called the method of Lagrange multipliers. up down return end

  27. Example Find the maximum value of the function f(x, y, z)=x+2y+3z on the curve of intersection of the plane x–y+ z=1 and the cylinder x2+y2=1. (p826) up down return end

  28. 11.1Three-dimensional coordinate systems 1. Recall coordinate systems in plane which locate the points in plane by an ordered pair (a,b) of real number, where a is the x-coordinate and b is the y-coordinate. 2. We can use a similar idea to locate the points in space by an ordered triple (a,b,c). We first choose a fixed point O (called origin) and three axes(directed lines) through O that are perpendicular to each other, called the coordinate axes and labeled the x-axis, y-axis, and z-axis. up down return end

  29. The three coordinate planes divide space into eight parts, called octants. They are called the first, the second,... ,the eighth octant respectively, illustrated in the figure. z III II VI I yz-plane xz-plane y o IV xy-plane VIII VII V x x-axis and y-axis form a plane coordinate and the direction of z-axis is determined by the Right-hand rule up down return end

  30. For every point P in space, there is only an ordered triple (a,b,c) of real numbers by three planes which are parallel to three coordinate planes respectively through P. a,b,c are interceptes with three coordinate axes, respectively. z (0,0,c) (0,b,c) - c (a,0,c) P (a,b,c) - 1 1 b y + + o - 1 xy-plane a _ (a,b,0) (a,0,0) x This is called three-dimensional rectangular coordinate system up down return end

  31. z (0,b,0) o y b a (a,0,0) (a,b,0) - (0,0,c) - c x (a,b,c) Contrary to this, for every ordered triple (a,b,c), there is only one point P, common point of three planes which are perpendicular to three coordinate axes respectively and through (a,0,0), (0,b,0), (0,0,c), respectively. up down return end

  32. P2(x2, y2 , z2) z z2 P2 z1 P1 P1(x1, y1 , z1) A B y1 y2 x1 y x2 x 2. We now can get a distance formula: Let P1(x1, y1 , z1) and P2 (x2 , y2 , z2 ) be the points in space, Then the distance | P1P2| between the two points is | P1P2|= Proof: up down return end

  33. 3. Example Find the distance from P(2,-1,7) to Q(1,-3,5). 4. Find an equation of a sphere with radius r and center C(h,k,l). SOLUTION: Let P(x, y, z) be a point in the sphere. So the distance |PC|=r. Then we can easily obtain that (x- h) 2+ (y- h) 2 + (z- h)2= r2. 5. From 4, every point in the sphere, the coordinate of the point must satisfies the Equation. On other hand, every ordered triple (x, y, z) which satisfies the Equation above, the point that is determined by (x, y, z) in the space must be on the sphere with radius r and center C(h,k,l). 6. Generally very equation F(x, y, z) =0 with three variables (x, y, z) represents a surface in space and vice versa. up down return end

  34. 7. Example: Discribe the surface which corresponds the equation x 2+ y 2 + z2 +4x-6y-8z= 5. 8. Equation Ax + By +Cz +D=0 (where A,B,C,D are constants) represents a plane in three-dimensional space. For instance, x=0 is yz-plane , z=5 is the plane parallel to and 5 units above xy-plane. How about y=10? 9. A special equation z=f(x, y), if (x, y) is confined in a domain D, means a piece of surface over or below the domain D. up down return end

  35. z (x, y, f(x,y)) o y D x (x,y) up down return end

  36. 10. The curve is intersection of two surfaces F(x,y,z)=0 and G(x,y,z)=0. 11. The intersection of two planes A1x + B1y +C1z +D1=0 and A2x + B2y +C2z +D2=0 is a straight line,or line for short. 12. Similarly curve in space also can be represented by parameter equation: x=f(t), y=g(t), z=h(t). 13. Similarly curve in space also can be represented by parameter equation: x=f(t), y=g(t), z=h(t). And if the curve: x=f(t), y=g(t), z=h(t) (atb) is smooth, then length of the curve is up down return end

  37. z 0 0 0 0. 0 0 0 0 0 0 0 0 0 0 y x 11.2 Quadratic surface (1) Ellipsoid: The quadratic surface with equation is calledellipsoid. up down return end

  38. (2) Hyperboloid of one sheet: z y H H H H H H x up down return end

  39. (3) Hyperboloid of two sheets: up down return end

  40. (4) Cones: z y H H H H H H x up down return end

  41. (5) Elliptic paraboloid: The surface Fig. up down return end

  42. (6) Hyperbolic paraboloid: The surface Fig. up down return end

  43. (7) Elliptic cylinder: The surface z y x2+y2=R2 x up down return end

  44. (8) Parabolic cylinder: The surface z y o y=x2 x up down return end

  45. 11.3(chapter 12.1) functions of several variables 1. Definition Let DR2. A function f of two variables is a rule that assigns to each ordered pair (x,y) in 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 value that f takes on, that is ,{f (x,y) | (x,y)D}. From the definition,we write z= f (x, y) to make explicit the value taken by f at general point (x,y). The variables x and y are independent variables and z is the dependent variable. Here D is represented as a subset of the xy- plane.Actually z= f (x, y) is a surface in space. up down return end

  46. 2.ExampleFind the domain of the functionf (x,y)=xln( y2-x) and evaluate f (3,2). SolutionSincef (x, y)=xln( y2-x) is defined only when y2-x >0,that is , y2>x ,the domain of f is D={(x,y)| y2>x }.This is a set of points to the left of parabola y2=x . And f (3,2)=3ln(22-3)=0 up down return end

  47. z A(0,0,1) C(0,0.25,0) o B(0.5,0,0) y x 3. Definition If is a function f of two variables with domain D, the graph of f is the set: S={(x,y,z)R3 | z=f (x,y) , (x,y)D}. Actually graph of a function of two variables is a surface. For instance, z= -2x-4y+1 is a plane through A(0,0,1), B(0.5,0,0),and C(0,0.25,0). up down return end

  48. Tell what are the surfaces of the following functions: up down return end

  49. 4. DefinitionThe level curve of a function f of two variables is the curve with equation f( x,y) =k, where k is a constant (in the range of f). Example is upper ellipsoid, whose domain is ellipse From the definition,we know that the level curve of a function f of two variables is vertical projection to xy-plane of intersection of surface z = f( x,y) and plane z =k, where k is a constant (in the range of f). up down return end

  50. 11.4 (chapter 12.2) Limit and continuity f(x,y) L as (x,y) (a,b). or 1. DefinitionLet f be a function of two variables defined a domain which contains a disk with center (a,b). Then we say that the limit of f (x, y) as (x, y) approaches (a, b) is L and we write or If for every number>0 there is a corresponding number >0 such that |f (x,y) - L |<  whenever up down return end

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