Mathematical analysis limit, continuous, derivable, differentiable, the presence of the continuous partial derivatives, the presence of mixed partial derivative with continuous, definition and the relationship directional derivative finishing
One centralized:
1 . Limit:
1) limit the number of columns:
Provided $ {a_n} $ several columns, a is the given number if positive for any given $ \ varepsilon $, there is always a positive integer N, such that when n> N when there is \ [{|. A_ {n} -a | } <{\ varepsilon}, \] called series $ {a_ {n}} $ converges to a, a set number of a called number sequence $ {a_n} $ limit, and recorded as $ \ lim_ {n \ to \ infty} {a_n} = a $, or $ a_n \ to a (a \ to \ infty) $, \\ read "when n goes to infinity, $ {a_n} $ a limit equal to or become a $ $ A_N . "
2) functional limit:
Defined. 1 ($ \ $ infty defined): Let f defined in $ [a, \ infty) function on $, A is a constant number. There \ [{if for any given $ {\ varepsilon}> {0 } $, there is a positive number $ M ({M} \ ge {a}) $, such that when the $ x> M $ | f ( x) -A |} <{\ varepsilon} \], when the called function $ f $ X $ $ $ + tends to \ infty $ when the limit to $ A $, denoted as \ [\ lim_ {x \ to + \ infty } = A or f (x) = A (x \ to + \ infty) \]
Definition 2: set function $ f $ at a point $ x_0 $ of a hollow art $ \ mathring {U} (x_0 ; \ delta ^ ') has $ within the definition, $ A $ is a given number if $ for any given \. vareplison> 0 $, there is a positive number $ \ delta (\ delta <\ delta ^ ') $, such that when the $ 0 <| x-x_0 | <\ when Delta $ there \ [{| f (x) -A |} < {\ varepsilon}, \] function called when $ f $ X $ $ $ tends to $ A $ $ x_0 the limit, referred to as \ [\ lim_ {x \ to x_0} {f (x)} = A or f (x) \ to A ( x \ to x_0). \]
Due principle (Theorem Heine) : $ \ lim_ {X \ to x_0} {F (X)} = A \ Longleftrightarrow any $ $ x_n \ to x_0 (n \ to \ infty) $ has $ \ lim_ {n \ to \ infty} {f (x_n )} = A. $
2. Continuous : $ f $ function set is defined in a $ U (x_0) $. If \ [\ lim_ {x \ toat a point called f $ $ x_0 continuous.
It is also equivalent to $ \ lim _ {\ Delta x \ to \ infty} \ Delta y = 0. $
Also be $ "\ varepsilon- \ delta" $ description language, i.e. for any given $ \ varepsilon> 0 $, the presence of $ \ delta> 0 $, such as $ | x-x_0 | <\ $ when there delta $ { . | f (x) -f ( x_0) |} <{\ varepsilon} $, $ f $ said function at a point $ $ x_0 continuous
Note: the "f at a point $ $ x_0 continuous" means that the corresponding law limit operation i.e. interchangeability: $ \ lim_ [X \ x_0 to] {F (X)} = F (\ lim_ {X \ x_0} to {X}) $
3. derivative: provided the function $ y = f (x) x_0 $ $ $ at the point of a neighborhood is defined, if the limit \ [lim_ {x \ to x_0 } frac {f (x) -f (x_0)} {x-x_0} \] exists, called function $ f x_0 $ $ $ at the point may be turned, and said limit is a function of the point $ f $ $ $ derivative at x_0, denoted F $ '(x_0) $.
4. differential: provided the function $ y = f (x ) $ defined in $ x_ {0} $ $ on a neighborhood $ U (x_0). when a $ x_ {0} $ increment $ \ delta {x}, x_ {0} + \ delta {x} \ in {U (x_ {0} )} $ , the corresponding function is obtained increment
\ [\ Delta} {y = f (x_ {0} + \ x {Delta} +) - f (x_ {0} \]
If the constant presence of $ A $, so $ \ Delta {y} $ can be expressed as \ [\ Delta {y} = A \ Delta {x} + o (\ Delta {x}), \] $ f $ function called at point $ x_ {0} $ differentiable, adding $ a \ Delta {x} $ point is $ f $ $ x_ {0} at $ differential, referred to as $ dy | _ {x = x_ {0} } = A \ Delta {x} $ or $ df (x) | _ { x = x_ {0}} = A \ Dealta {x} $
continuous (limit exists and is equal to the function value) $ \ Longlleftarrow $ derivable (defined limit formula present) $ \ Longleftrightarrow $ differentiable
Second, multivariate function (a binary Example)
1. Limit:
Definition 1: provided as defined in $ f $ $ {D} \ subset {R ^ {2}} $ on a binary function, $ P_ {0} $ $ D $ is an accumulation point, $ A $ is a determining the real if for any given integer $ \ varepsilon $, there is always some integer $ \ delta $, such as $ {P} in {{\ mathring {U} (P_0; \ delta)} \ cap {D}} when $, both \ [{| f (P) -A |} <{\ varepsilon} \], called $ f $ taken in $ D $ $ P \ to {P_0} $ when the limit to $ A $ , denoted as: \ [\ lim _ {{ P} \ to {P_0} \\ P \ in {D}} {f (P)} = A. \]
When the two-dimensional coordinates is expressed as $ \ lim _ {(x, y) \ to (x_0, y_0)} {f (x, y)} = A. $
Definition 2: Let D be the domain of binary function f, $ P_0 (x_0, y_0) $ point D is a polyethylene. If the positive number M of any given, there is always a $ site $ P_0 $ a \ Delta $ neighborhood, such as $ P (x, y) \ in {{{\ mathring {U} (P_0; \ delta)} \ cap {D}} $ when there $ f (P)> M $ , $ f $ claimed in $ D $ taken $ P \ to {P_0} $, there is a non-normal limit of $ + \ infty $, denoted as $ \ lim _ {(x, y) \ to (x_0, y_0)} f (x, y) = + \ infty. $
Or $ \ lim_ {p \ to { P_0}} f (P) = + \ infty $.
Above discussion in any way limit close to a specific point, referred to the weight limit .
Repeated Limit:
Definition: set $ f (x , y), (x, y ) \ in {D}, D $ $ X $ axis in the projection $ $ Y axis respectively $ X, Y $, ie \ [X = {x | ( x, y ) \ in {D}}, Y = {y | (x, y) \ in {D}}, \] $ x_0, y_0 $ are $ X, Y $ accumulation point, if for every $ Y \ in {Y}, $ there is a limit $ \ lim_ {x \ to { x_0}} f (x, y), $ which is generally $ Y $ related, it referred to as \ [\ varphi (y) = \ lim_ {x \ to {x_0]} f ( x, y), \]
If there is further limit \ [L = \ lim_ {y \ to {y_0}} \ varphi (y), \]
This limit is called L $ f (x, y) $ firstly $ x (\ to {x_0}) after $ to $ y (to {y_0}) $ Double Limit, referred to as \ [L = \ lim_ {y \ to {y_0}} \ lim_ {x \ to {x_0}} f (x, y). \]
Similar defined $ K = \ lim_ {x \ to {x_0]} \ lim_ {y \ to}} f (x, y) $ {y_0.
Relationship between the weight limit and Repeated Limit :
A point $ P_0 (x_0, y_0) $ Repeated limit and weight limit exists (to one), then they will be equal;
$\Longrightarrow $
1. Repeated limit, there is a weight limit is equal to three whoever
2. Repeated two limits exist but will not equal the weight limit does not exist.
Example 1: $ f (x, y ) = frac {xy} {x ^ 2 + y ^ 2} $.
Example 2: $ f (x , y) = frac {x- y + x ^ 2 + y ^ 2} {x + y} $.
Example 3: $ f (x, y ) = x \ sin \ frac {1} {y} + y \ sin \ frac {1} {x }. $
2. Continuous:
Definition: The set point for defining $ f $ set $ D \ subset {R ^ 2 } $ on a binary function, $ P_0 \ in {D} $ ($ P_0 $ poly isolated dots or points) to any given in positive $ \ varepsilon $, there is always a positive number corresponding $ \ delta $, as long as the $ P \ in {U (P_0 ; \ delta) \ cap {D}}, $ there \ [| f (P) -f ( P_0) | <\ varepsilon, \ ] \
Claimed that $ f $ on the set $ D $ at the point $ P_0 $ continuous.
Incremental form: \ [\ lim _ {( \ Delta {x}, \ Delta {y}) \ to (0,0) \] \ [(x, y) \ in {D}} \ Delta {z} = 0. \] $ \ Longrightarrow $ $ f $ $ P_0 $ point continuous.
3. differentiable: Definition: provided the function $ z = f (x, y ) $ o at a point $ P_0 (x_0, y_0) $ of domain $ U (P_0) is defined $ on, for $ U (P_0) $ a point $ P (x, y) = (x_0 + \ Delta {x}, y_0 + \ Delta {y}) $, if the function f point $ P_ {0} full incremental $ at \ delta {z} may be represented as \ [\ begin {align *} \ delta {z} & = f (x_0 + \ delta {x}, y_0 + \ delta {y} ) -f (x_0, y_0) \\ & = A \ Delta {x} + B \ Delta {y} + o (\ rho) \\ & = A \ Delta {x} + B \ Delta {y} + \ alpha \ Delta {x} + \ beta \ Delta {y} \\ \ lim _ {(\ Delta {x}, \ Delta {y}) \ to (0,0)} \ alpha = \ lim _ {(\ Delta { x}, \ Delta {y} ) \ to (0,0)} \ beta = 0. \ end {align *} \]
wherein a, B is associated with only a point $ $ P_0 constant, $ \ rho = sqrt {\ Delta {x} ^ 2 + \ Delta {y} ^ 2} $ \ qquad, o (\ rho) is less $ \ $ Rho-order infinitely small, the function f may be called a micro point $ P_0 $. and said about $ \ Delta {x}, \ Delta {y} $ linear function $ a \ Delta {x} + B \ Delta {y} $ $ f $ as a function of the total differential P_0 point $ $, denoted by $$ dz | _ {P_0} = df (x_0, y_0) = A \ Delta {x} + B \ Delta {y} $$
4. The partial derivatives: provided the function $ z = f (x, yIf $ (x_0, y_0) \ inand $ f (x, y_0. ) $ at $ x_0 $ of a neighborhood is defined, then when the limit \ [\ lim _ {\ Deltax}} = \ lim _ {exists when, called this limit is a function of $ f $ at the point $ (x_0, y_0) $ partial derivative with respect to $ x $, and
Referred to as $ f_ {x} (x_0, y_0) $ or $ z_ {x} (x_0, y_0), \ frac {\ partial f} {\ partial x} | _ (x_0, y_0), \ frac {\ partial z} {\ partial x} _ (x_0, y_0). $
continuous \ presence $$ longrightarrow differentiable \ longleftrightarrow partial derivative $$ partial derivative
5. direction 导数:
Definition: ary function is provided at a point $ f $ $ P_ {0} (x_0, y_0, z_0) $ certain neighborhood $ U (P_0) \ subset { R ^ 3} $ is defined, L $ $ from point $ P_ {0} $ starting rays, $ P (x, y, z) $ is the $ L $ and included in $ U (P_0) any point $ in order $ \ Rho $ represents $ P $ and $ the distance between P_0 $ two points if the limit \ [\ lim _ {\ rho \ to {0 ^ +}} \ frac {f (P) -f (P_0)} {\ rho} = \ lim _ {\ rho \ to {0 ^ +}} \ frac {\ delta_ {l} f} {\ rho} \] exists, called this limit as a function of $ f $ in the directional derivative point $ P_0 $ direction $ L $, denoted \ [frac {partical f} {partical l} |. _ {P_0}, f_ {l} (P_0) or f_ {l} (x_0, y_0 , z_0) \]
If $ f $ in $ P_0 $ differentiable, then the directional derivative in $ P_04 either direction $ L $ are present, and $ f_ {l} (P_0) = f_ {x} (P_0) \ cos {\ alpha} + f_ {y} (P_0) \ cos { \ beta} + f_ {z} (P_0) \ cos {\ gamma}, $ where $ \ cos \ alpha, \ cos \ beta, \ cos \ gamma $ a direction $ cosine L $ direction.
introduction gradient (gradient) : $ grad = (f_ {x } (P_0), f_ {y} (P_0), f_ {z} (P_0)) $.
$ And $ L units may be turned into $ l_0 = (\ cos \ alpha, \ cos \ beta, \ cos \ gamma) $
Further F_ {L} $ (P_0) = gradf (P_0) \ CDOT L_0} = {| gradf (P_0) | \ COS \ Theta $
6. The mixed partial derivative will be continuously equal. (+ Constructor value theorem)