Khác biệt giữa các bản “Đạo hàm riêng”

:<math>\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(x_i) = \frac{\part f}{\part x_i}(a_1,\ldots,a_n).</math>
 
In other words, the different choices of ''a'' index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.
 
AnMột important exampledụ ofquan atrọng functioncủa ofđạo severalhàm variablesriêng: isCho the case of amột [[scalar-valuedhàm function]]số ''f''(''x''<sub>1</sub>,...''x''<sub>''n''</sub>) onđinh anghĩa domaintrên inmột Euclideanmiền spacecủa '''R'''<sup>''n''</sup> (e.g.ví dụ, ontrên '''R'''<sup>2</sup> orhay là '''R'''<sup>3</sup>). InTrong trường thishợp casenày L''f'' has acác đạo partialhàm derivativeriêng ∂''f''/∂''x''<sub>''j''</sub> withđối respectvới tomỗi each variablebiến ''x''<sub>''j''</sub>. AtTại theđiểm point ''a'', thesenhững đạo hàm partialriêng derivativesnày defineđịnh thera vector
 
:<math>\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).</math>
 
ThisVector vectornày isđược calledgọi the '''[[gradient]]''' ofcủa ''f'' attại ''a''. IfNếu ''f'' iskhả differentiablevi attại everymọi pointđiểm introng somemột domain,miền thennào đó, thethì gradient is ahàm số có vector-valuedtrị functionlà vectơ ∇''f'' whichđưa takes the pointđiểm ''a'' tođến the vectorvectơ ∇''f''(''a''). Consequently,Do theđó gradient produces amột [[vector field]].
[[trường vectơ]].
 
A common [[abuse of notation]] is to define the [[del operator]] (∇) as follows in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup> with [[unit vectors]] <math>\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}</math>:
:<math>\nabla = \bigg[{\frac{\partial}{\partial x}} \bigg] \mathbf{\hat{i}} + \bigg[{\frac{\partial}{\partial y}}\bigg] \mathbf{\hat{j}} + \bigg[{\frac{\partial}{\partial z}}\bigg] \mathbf{\hat{k}}</math>
Or, more generally, for ''n''-dimensional Euclidean space '''R'''<sup>''n''</sup> with coordinates (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>,...,x<sub>''n''</sub>) and unit vectors (<math>\mathbf{\hat{e}_1}, \mathbf{\hat{e}_2}, \mathbf{\hat{e}_3}, \dots , \mathbf{\hat{e}_n}</math>):
:<math>\nabla = \sum_{j=1}^n \bigg[{\frac{\partial}{\partial x_j}}\bigg] \mathbf{\hat{e}_j} = \bigg[{\frac{\partial}{\partial x_1}}\bigg] \mathbf{\hat{e}_1} + \bigg[{\frac{\partial}{\partial x_2}}\bigg] \mathbf{\hat{e}_2} + \bigg[{\frac{\partial}{\partial x_3}}\bigg] \mathbf{\hat{e}_3} + \dots + \bigg[{\frac{\partial}{\partial x_n}}\bigg] \mathbf{\hat{e}_n}</math>
 
==Examples==
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