Download Calculus III: Functions of Several Variables, Limits, and Partial Derivatives - Ch. 14-15 and more Exams Advanced Calculus in PDF only on Docsity!
MATH 2433, Calculus III
Summary: CHAPTERS 14 and 15
- FUNCTIONS OF SEVERAL VARIABLES: Let z = f (x, y) or w = f (x, y, z).
Domain D: If the domain of f is not given explicitly, or implicitly by an application, then, by convention, the domain is the set of all points x [x = (x, y) or x = (x, y, z)] such that f (x) is a real number. Range: The range of f is the set of values f (x), x ∈ D. Graph: Let z = f (x, y), (x, y) ∈ D. The graph of f is the set of all points (x, y, z) = (x, y, f (x, y)) in space. For our purposes, the graph of z = f (x, y) is a surface in space. The graph of w = f (x, y, z) is a “hypersurface” in 4-dimensional space. Level curves and level surfaces: Given z = f (x, y). The plane curves
f (x, y) = C, C constant
are called the level curves of f. Let w = F (x, y, z). The surfaces in 3-space
F (x, y, z) = C, C constant
are called the level surfaces of F.
- LIMITS: Given z = f (x, y) or w = f (x, y, z).
Let x 0 = (x 0 , y 0 ) or (x 0 , y 0 , z 0 ) and x = (x, y) or (x, y, z). Assume that f is defined in some neighborhood of the point x 0 except, possibly, at x 0 itself.
xlim→x 0
f (x) = L
if for each > 0 there exists a δ > 0 such that
|f (x) − L| < whenever ‖x − x 0 ‖ < δ.
- PARTIAL DERIVATIVES: Given a function z = f (x, y). Let x = (x, y).
The partial derivative of f with respect to x at the point x is given by
fx = lim h→ 0
f (x + h, y) − f (x, y) h
= lim h→ 0
f (x + h i) − f (x) h
(provided the limit exists).
The partial derivative of f with respect to y at the point x is given by
fy = lim h→ 0
f (x, y + h) − f (x, y) h
= lim h→ 0
f (x + h j) − f (x) h
(provided the limit exists).
Corresponding definitions hold for the function w = F (x, y, z). For example: The partial derivative of F with respect to z at the point x = (x, y, z) is given by
Fz = lim h→ 0
F (x, y, z + h) − F (x, y, z) h
= lim h→ 0
F (x + h k) − F (x) h
(provided the limit exists).
Notations: fx =
∂f ∂x
∂z ∂x
, fy =
∂f ∂y
∂z ∂y
, etc.
Higher Derivatives:
fxx =
∂^2 f ∂x^2
∂ (∂f /∂x) ∂x
, fxy =
∂^2 f ∂y ∂x
∂ (∂f /∂x) ∂y
fyy =
∂^2 f ∂y^2
∂ (∂f /∂y) ∂y
, fyx =
∂^2 f ∂x ∂y
∂ (∂f /∂y) ∂x
THEOREM: If the first partials and the “mixed” second partials are continuous on D, then fxy = fyx.
- GRADIENT: Given z = f (x, y); w = F (x, y, z)
Gradient of f : ∇f = fx i + fy j gradient of F : ∇F = Fx i + Fy j + Fz k ∇f is normal to the level curves of f ; ∇F is normal to the level surfaces of F.
Directional derivatives: Let u = u 1 i + u 2 j be a unit vector and let x = (x, y). The directional derivative of f at x in the direction u is given by:
f (^) u′ = lim h→ 0
f (x + h u) − f (x) h
provided the limit exists
Similarly, if u = u 1 i+u 2 j+u 3 k is a unit vector and x = (x, y, z), then the directional derivative of F at x in the direction u is
F (^) u′ = lim h→ 0
F (x + h u) − F (x) h
provided the limit exists
Calculation of directional derivatives:
f (^) u′ = ∇f (x)·u; F (^) u′ = ∇F (x)·u.
Tangent planes and normal lines: Equations for the tangent plane and normal line to the surface z = f (x, y) at the point (x 0 , y 0 , z 0 ), [z 0 = f (x 0 , y 0 )] are given by:
tangent plane: fx(x 0 , y 0 )(x − x 0 ) + fy (x 0 , y 0 )(y − y 0 ) − (z − z 0 ) = 0; normal line: x = x 0 + fx(x 0 , y 0 ) t, y = y 0 + fy (x 0 , y 0 ) t, z = z 0 − t
- ABSOLUTE EXTREME VALUES: Let f be a function of several variables [z = f (x, y), w = f (x, y, z)], and let x 0 be a point in D, the domain of f.
(a) f has an absolute maximum at x 0 if f (x 0 ) ≥ f (x) for all x in D.
(b) f has aa absolute minimum at x 0 if f (x 0 ) ≤ f (x) for all x in D.
(c) Theorem: If f is continuous and D is closed and bounded, then f has an absolute maximum and an absolute minimum on D. These occur either at a critical point in the interior of D or at a point on the boundary of D. [C.f., a continuous function on a closed and bounded interval, see Sections 2.6 and 4.4.]
8. LAGRANGE MULTIPLIERS:
If x 0 maximizes or minimizes f subject to the side condition g(x) = c, then there exists a scalar λ such that
∇f (x 0 ) = λ ∇g(x 0 ) (provided ∇g(x 0 ) 6 = 0).
To find the maxima/minima of z = f (x, y) subject to the side condition g(x, y) = c, solve the system of equations:
fx(x, y) = λ gx(x, y) fy (x, y) = λ gy(x, y) g(x, y) = c
To find the maxima/minima of w = F (x, y, z) subject to the side condition G(x, y, z) = c, solve the system of equations:
Fx(x, y, z) = λ Gx(x, y, z) Fy (x, y, z) = λ Gy(x, y, z) Fz (x, y, z) = λ Gz(x, y, z) G(x, y, z) = c
9. RECONSTRUCTING A FUNCTION FROM ITS GRADIENT:
Let P = P (x, y) and Q = Q(x, y) be continuously differentiable functions on a simply connected open region Ω. The vector function R(x, y) = P (x, y) i + Q(x, y) j is the gradient of some function z = f (x, y) on Ω if and only if
∂P ∂y
∂Q
∂x
If R is the gradient of a function f , then
f (x, y) =
∫ P (x, y) dx + φ(y)
where φ is a function of y which is to be determined so that ∂f /∂y = Q(x, y). Alternatively,
f (x, y) =
∫ Q(x, y) dy + ψ(y)
where ψ is a function of x which is to be determined so that ∂f /∂x = P (x, y).
