上面兩個圖中，純量場的值用灰度 表示，越暗表示越大的數值，而其相應的梯度用藍色箭頭表示。 在向量微積分 中，梯度 （英語：gradient ）是一種關於多元導數 的概括[1] 函數 的導數是純量值函數 ，而多元函數 的梯度是向量值函數 。多元 可微函數
f
{\displaystyle f}
P
{\displaystyle P}
f
{\displaystyle f}
P
{\displaystyle P}
偏導數 為分量的向量 [2]
就像一元函數的導數表示這個函數圖形 的切線 的斜率 [3]
P
{\displaystyle P}
方向 是這個函數在
P
{\displaystyle P}
量 是在這個方向上的增長率[4]
梯度向量中的幅值和方向是與坐標的選擇無關的獨立量[5]
在歐幾里德空間 或更一般的流形 之間的多元可微映射 的向量值函數的梯度推廣是雅可比矩陣 [6] 巴拿赫空間 之間的函數的進一步推廣是弗雷歇導數 。
梯度的解釋 將2D函數f (x , y ) = xe −(x 2 + y 2 ) 假設有一個房間，房間內所有點的溫度由一個純量場
ϕ
{\displaystyle \phi }
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
ϕ
(
x
,
y
,
z
)
{\displaystyle \phi (x,y,z)}
考慮一座高度在
(
x
,
y
)
{\displaystyle (x,y)}
H
(
x
,
y
)
{\displaystyle H(x,y)}
H
{\displaystyle H}
坡度 （或者說斜度 ）最陡的方向。梯度的大小告訴我們坡度到底有多陡。
梯度也可以告訴我們一個數量在不是最快變化方向的其他方向的變化速度。再次考慮山坡的例子。可以有條直接上山的路其坡度是最大的，則其坡度是梯度的大小。也可以有一條和上坡方向成一個角度的路，例如投影在水平面上的夾角為60°。則，若最陡的坡度是40%，這條路的坡度小一點，是20%，也就是40%乘以60°的餘弦。
這個現象可以如下數學的表示。山的高度函數
H
{\displaystyle H}
點積 一個單位向量 給出表面在該向量的方向上的斜率。這稱為方向導數 。
定義 將函數f (x ,y ) = −(cos2 x + cos2 y )2 向量場 。 純量函數
f
:
R
n
↦
R
{\displaystyle f\colon \mathbb {R} ^{n}\mapsto \mathbb {R} }
∇
f
{\displaystyle \nabla f}
grad
f
{\displaystyle \operatorname {grad} f}
∇
{\displaystyle \nabla }
nabla ）表示向量微分算子 。
函數
f
{\displaystyle f}
∇
f
{\displaystyle \nabla f}
v
(
∇
f
(
x
)
)
⋅
v
=
D
v
f
(
x
)
{\displaystyle {\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x)}
直角坐標系
∇
f
{\displaystyle \nabla f}
直角坐標系 中表示為
∇
f
=
(
∂
f
∂
x
,
∂
f
∂
y
,
∂
f
∂
z
)
=
∂
f
∂
x
i
+
∂
f
∂
y
j
+
∂
f
∂
z
k
{\displaystyle \nabla f={\begin{pmatrix}{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\end{pmatrix}}={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }
i j k x y z 參看偏導數 和向量 。 ）
雖然使用坐標表達，但結果是在正交轉換 下不變，從幾何的觀點來看，這是應該的。
舉例來講，函數
f
(
x
,
y
,
z
)
=
2
x
+
3
y
2
−
sin
(
z
)
{\displaystyle f(x,y,z)=2x+3y^{2}-\sin(z)}
∇
f
=
(
2
,
6
y
,
−
cos
(
z
)
)
=
2
i
+
6
y
j
−
cos
(
z
)
k
{\displaystyle \nabla f={\begin{pmatrix}{2},{6y},{-\cos(z)}\end{pmatrix}}=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} }
圓柱坐標系 在圓柱坐標系 中，
f
{\displaystyle f}
[7]
∇
f
(
ρ
,
φ
,
z
)
=
∂
f
∂
ρ
e
ρ
+
1
ρ
∂
f
∂
φ
e
φ
+
∂
f
∂
z
e
z
{\displaystyle \nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z}}
ρ φ 投影線 與正 x-軸之間的夾角。
z 直角坐標 的
z
{\displaystyle z}
e ρ e φ e z
球坐標系 在球坐標系 中：
∇
f
(
r
,
θ
,
φ
)
=
∂
f
∂
r
e
r
+
1
r
∂
f
∂
θ
e
θ
+
1
r
sin
θ
∂
f
∂
φ
e
φ
{\displaystyle \nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }}
其中θ φ
實值函數相對於向量和矩陣的梯度 相對於n×1向量x 的梯度算子記作
∇
x
{\displaystyle \nabla _{\boldsymbol {x}}}
[8]
∇
x
=
d
e
f
[
∂
∂
x
1
,
∂
∂
x
2
,
⋯
,
∂
∂
x
n
]
T
=
∂
∂
x
{\displaystyle \nabla _{\boldsymbol {x}}{\overset {\underset {\mathrm {def} }{}}{=}}\left[{\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},\cdots ,{\frac {\partial }{\partial x_{n}}}\right]^{T}={\frac {\partial }{\partial {\boldsymbol {x}}}}}
對向量的梯度 以n×1實向量x 為變元的實純量函數f(x )相對於x 的梯度為一n×1列向量x ，定義為
∇
x
f
(
x
)
=
d
e
f
[
∂
f
(
x
)
∂
x
1
,
∂
f
(
x
)
∂
x
2
,
⋯
,
∂
f
(
x
)
∂
x
n
]
T
=
∂
f
(
x
)
∂
x
{\displaystyle \nabla _{\boldsymbol {x}}f({\boldsymbol {x}}){\overset {\underset {\mathrm {def} }{}}{=}}\left[{\frac {\partial f({\boldsymbol {x}})}{\partial x_{1}}},{\frac {\partial f({\boldsymbol {x}})}{\partial x_{2}}},\cdots ,{\frac {\partial f({\boldsymbol {x}})}{\partial x_{n}}}\right]^{T}={\frac {\partial f({\boldsymbol {x}})}{\partial {\boldsymbol {x}}}}}
m維行向量函數
f
(
x
)
=
[
f
1
(
x
)
,
f
2
(
x
)
,
⋯
,
f
m
(
x
)
]
{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=[f_{1}({\boldsymbol {x}}),f_{2}({\boldsymbol {x}}),\cdots ,f_{m}({\boldsymbol {x}})]}
x 的梯度為一n×m矩陣，定義為
∇
x
f
(
x
)
=
d
e
f
[
∂
f
1
(
x
)
∂
x
1
∂
f
2
(
x
)
∂
x
1
⋯
∂
f
m
(
x
)
∂
x
1
∂
f
1
(
x
)
∂
x
2
∂
f
2
(
x
)
∂
x
2
⋯
∂
f
m
(
x
)
∂
x
2
⋮
⋮
⋱
⋮
∂
f
1
(
x
)
∂
x
n
∂
f
2
(
x
)
∂
x
n
⋯
∂
f
m
(
x
)
∂
x
n
]
=
∂
f
(
x
)
∂
x
{\displaystyle \nabla _{\boldsymbol {x}}{\boldsymbol {f}}({\boldsymbol {x}}){\overset {\underset {\mathrm {def} }{}}{=}}{\begin{bmatrix}{\frac {\partial f_{1}({\boldsymbol {x}})}{\partial x_{1}}}&{\frac {\partial f_{2}({\boldsymbol {x}})}{\partial x_{1}}}&\cdots &{\frac {\partial f_{m}({\boldsymbol {x}})}{\partial x_{1}}}\\{\frac {\partial f_{1}({\boldsymbol {x}})}{\partial x_{2}}}&{\frac {\partial f_{2}({\boldsymbol {x}})}{\partial x_{2}}}&\cdots &{\frac {\partial f_{m}({\boldsymbol {x}})}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f_{1}({\boldsymbol {x}})}{\partial x_{n}}}&{\frac {\partial f_{2}({\boldsymbol {x}})}{\partial x_{n}}}&\cdots &{\frac {\partial f_{m}({\boldsymbol {x}})}{\partial x_{n}}}\\\end{bmatrix}}={\frac {\partial {\boldsymbol {f}}({\boldsymbol {x}})}{\partial {\boldsymbol {x}}}}}
對矩陣的梯度 純量函數
f
(
A
)
{\displaystyle f({\boldsymbol {A}})}
A 的梯度為一m×n矩陣，簡稱梯度矩陣，定義為
∇
A
f
(
A
)
=
d
e
f
[
∂
f
(
A
)
∂
a
11
∂
f
(
A
)
∂
a
12
⋯
∂
f
(
A
)
∂
a
1
n
∂
f
(
A
)
∂
a
21
∂
f
(
A
)
∂
a
22
⋯
∂
f
(
A
)
∂
a
2
n
⋮
⋮
⋱
⋮
∂
f
(
A
)
∂
a
m
1
∂
f
(
A
)
∂
a
m
2
⋯
∂
f
(
A
)
∂
a
m
n
]
=
∂
f
(
A
)
∂
A
{\displaystyle \nabla _{\boldsymbol {A}}f({\boldsymbol {A}}){\overset {\underset {\mathrm {def} }{}}{=}}{\begin{bmatrix}{\frac {\partial f({\boldsymbol {A}})}{\partial a_{11}}}&{\frac {\partial f({\boldsymbol {A}})}{\partial a_{12}}}&\cdots &{\frac {\partial f({\boldsymbol {A}})}{\partial a_{1n}}}\\{\frac {\partial f({\boldsymbol {A}})}{\partial a_{21}}}&{\frac {\partial f({\boldsymbol {A}})}{\partial a_{22}}}&\cdots &{\frac {\partial f({\boldsymbol {A}})}{\partial a_{2n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f({\boldsymbol {A}})}{\partial a_{m1}}}&{\frac {\partial f({\boldsymbol {A}})}{\partial a_{m2}}}&\cdots &{\frac {\partial f({\boldsymbol {A}})}{\partial a_{mn}}}\\\end{bmatrix}}={\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}
法則 以下法則適用於實純量函數對向量的梯度以及對矩陣的梯度。
線性法則：若
f
(
A
)
{\displaystyle f({\boldsymbol {A}})}
g
(
A
)
{\displaystyle g({\boldsymbol {A}})}
1 和c2 為實常數，則
∂
[
c
1
f
(
A
)
+
c
2
g
(
A
)
]
∂
A
=
c
1
∂
f
(
A
)
∂
A
+
c
2
∂
g
(
A
)
∂
A
{\displaystyle {\frac {\partial [c_{1}f({\boldsymbol {A}})+c_{2}g({\boldsymbol {A}})]}{\partial {\boldsymbol {A}}}}=c_{1}{\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}+c_{2}{\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}
乘積法則：若
f
(
A
)
{\displaystyle f({\boldsymbol {A}})}
g
(
A
)
{\displaystyle g({\boldsymbol {A}})}
h
(
A
)
{\displaystyle h({\boldsymbol {A}})}
∂
f
(
A
)
g
(
A
)
∂
A
=
g
(
A
)
∂
f
(
A
)
∂
A
+
f
(
A
)
∂
g
(
A
)
∂
A
{\displaystyle {\frac {\partial f({\boldsymbol {A}})g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}=g({\boldsymbol {A}}){\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}+f({\boldsymbol {A}}){\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}
∂
f
(
A
)
g
(
A
)
h
(
A
)
∂
A
=
g
(
A
)
h
(
A
)
∂
f
(
A
)
∂
A
+
f
(
A
)
h
(
A
)
∂
g
(
A
)
∂
A
+
f
(
A
)
g
(
A
)
∂
h
(
A
)
∂
A
{\displaystyle {\frac {\partial f({\boldsymbol {A}})g({\boldsymbol {A}})h({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}=g({\boldsymbol {A}})h({\boldsymbol {A}}){\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}+f({\boldsymbol {A}})h({\boldsymbol {A}}){\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}+f({\boldsymbol {A}})g({\boldsymbol {A}}){\frac {\partial h({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}
商法則：若
g
(
A
)
≠
0
{\displaystyle g({\boldsymbol {A}})\neq 0}
∂
f
(
A
)
/
g
(
A
)
∂
A
=
1
g
(
A
)
2
[
g
(
A
)
∂
f
(
A
)
∂
A
−
f
(
A
)
∂
g
(
A
)
∂
A
]
{\displaystyle {\frac {\partial f({\boldsymbol {A}})/g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}={\frac {1}{g({\boldsymbol {A}})^{2}}}\left[g({\boldsymbol {A}}){\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}-f({\boldsymbol {A}}){\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}\right]}
連鎖律：若A 為m×n矩陣，且
y
=
f
(
A
)
{\displaystyle y=f({\boldsymbol {A}})}
g
(
y
)
{\displaystyle g(y)}
A 和純量y為變元的實純量函數，則
∂
g
(
f
(
A
)
)
∂
A
=
d
g
(
y
)
d
y
∂
f
(
A
)
∂
A
{\displaystyle {\frac {\partial g(f({\boldsymbol {A}}))}{\partial {\boldsymbol {A}}}}={\frac {dg(y)}{dy}}{\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}
流形上的梯度 一個黎曼流形
M
{\displaystyle M}
f
{\displaystyle f}
∇
f
{\displaystyle \nabla f}
向量場 ，使得對於每個向量
ξ
{\displaystyle \xi }
⟨
∇
f
,
ξ
⟩
:=
ξ
f
{\displaystyle \langle \nabla f,\xi \rangle :=\xi f}
其中
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
M
{\displaystyle M}
內積 （度量）而
ξ
f
(
p
)
,
p
∈
M
{\displaystyle \xi f(p),p\in M}
f
{\displaystyle f}
p
{\displaystyle p}
ξ
(
p
)
{\displaystyle \xi (p)}
方向導數 。換句話說，如果
φ
:
U
⊆
M
↦
R
n
{\displaystyle \varphi :U\subseteq M\mapsto \mathbb {R} ^{n}}
p
{\displaystyle p}
ξ
(
x
)
=
∑
j
a
j
(
x
)
∂
∂
x
j
{\displaystyle \xi (x)=\sum _{j}a_{j}(x){\frac {\partial }{\partial x_{j}}}}
ξ
f
(
p
)
{\displaystyle \xi f(p)}
ξ
(
f
∣
p
)
:=
∑
j
a
j
(
∂
∂
x
j
(
f
∘
φ
−
1
)
∣
φ
(
p
)
)
{\displaystyle \xi (f\mid _{p}):=\sum _{j}a_{j}({\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1})\mid _{\varphi (p)})}
函數的梯度和外微分 相關，因為
ξ
f
=
d
f
(
ξ
)
{\displaystyle \xi f=df(\xi )}
d
f
{\displaystyle df}
∇
f
{\displaystyle \nabla f}
∇
f
{\displaystyle \nabla f}
d
f
(
ξ
)
=
⟨
∇
f
,
ξ
⟩
{\displaystyle df(\xi )=\langle \nabla f,\xi \rangle }
f
{\displaystyle f}
d
f
{\displaystyle df}
{
d
f
}
{\displaystyle \{df\}}
{
∇
f
}
{\displaystyle \{\nabla f\}}
滿射 。
由定義可算流形
∇
f
{\displaystyle \nabla f}
∇
f
=
∑
i
k
g
i
k
∂
f
∂
x
k
∂
∂
x
i
{\displaystyle \nabla f=\sum _{ik}g^{ik}{\frac {\partial f}{\partial x^{k}}}{\frac {\partial }{\partial x^{i}}}}
請注意這是流形
d
s
2
=
∑
i
j
g
i
j
d
x
i
d
x
j
{\displaystyle ds^{2}=\sum _{ij}g_{ij}dx^{i}dx^{j}}
R
n
{\displaystyle \mathbb {R} ^{n}}
∑
{\displaystyle \sum }
參看 參考文獻 引用
^ Beauregard & Fraleigh (1973 , p. 84)^ Bachman (2007 , p. 76)Beauregard & Fraleigh (1973 , p. 84)Downing (2010 , p. 316)Harper (1976 , p. 15)Kreyszig (1972 , p. 307)McGraw-Hill (2007 , p. 196)Moise (1967 , p. 683)Protter & Morrey, Jr. (1970 , p. 714)Swokowski et al. (1994 , p. 1038)^ Protter & Morrey, Jr. (1970 , pp. 21,88)^ Bachman (2007 , p. 77)Downing (2010 , pp. 316–317)Kreyszig (1972 , p. 309)McGraw-Hill (2007 , p. 196)Moise (1967 , p. 684)Protter & Morrey, Jr. (1970 , p. 715)Swokowski et al. (1994 , pp. 1036,1038–1039)^ Kreyszig (1972 , pp. 308–309)Stoker (1969 , p. 292)^ Beauregard & Fraleigh (1973 , pp. 87,248)Kreyszig (1972 , pp. 333,353,496)^ Schey 1992 ，第139–142頁.^ 張賢達 (2004 , p. 258)
來源
書籍 Bachman, David, Advanced Calculus Demystified, New York: McGraw-Hill , 2007, ISBN 0-07-148121-4 Beauregard, Raymond A.; Fraleigh, John B., A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company ISBN 0-395-14017-X Downing, Douglas, Ph.D., Barron's E-Z Calculus, New York: Barron's ISBN 978-0-7641-4461-5 Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. Modern Geometry—Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics 2nd. Springer. 1991. ISBN 978-0-387-97663-1 Harper, Charlie, Introduction to Mathematical Physics, New Jersey: Prentice-Hall , 1976, ISBN 0-13-487538-9 Kreyszig, Erwin , Advanced Engineering Mathematics 3rd, New York: Wiley , 1972, ISBN 0-471-50728-8 McGraw-Hill Encyclopedia of Science & Technology 10th. New York: McGraw-Hill . 2007. ISBN 0-07-144143-3 Moise, Edwin E., Calculus: Complete, Reading: Addison-Wesley , 1967 Protter, Murray H.; Morrey, Jr., Charles B., College Calculus with Analytic Geometry 2nd, Reading: Addison-Wesley , 1970, LCCN 76087042 Schey, H. M. Div, Grad, Curl, and All That 2nd. W. W. Norton. 1992. ISBN 0-393-96251-2OCLC 25048561 Stoker, J. J., Differential Geometry, New York: Wiley , 1969, ISBN 0-471-82825-4 Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A., Calculus 6th, Boston: PWS Publishing Company, 1994, ISBN 0-534-93624-5 張賢達 , 《矩阵分析与应用》, 清華大學出版社, 2004, ISBN 9787302092711（中文（中國大陸））
![{\displaystyle \nabla _{\boldsymbol {x}}{\overset {\underset {\mathrm {def} }{}}{=}}\left[{\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},\cdots ,{\frac {\partial }{\partial x_{n}}}\right]^{T}={\frac {\partial }{\partial {\boldsymbol {x}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038e398aeea589b3666797409a08dba0c541dc44)
![{\displaystyle \nabla _{\boldsymbol {x}}f({\boldsymbol {x}}){\overset {\underset {\mathrm {def} }{}}{=}}\left[{\frac {\partial f({\boldsymbol {x}})}{\partial x_{1}}},{\frac {\partial f({\boldsymbol {x}})}{\partial x_{2}}},\cdots ,{\frac {\partial f({\boldsymbol {x}})}{\partial x_{n}}}\right]^{T}={\frac {\partial f({\boldsymbol {x}})}{\partial {\boldsymbol {x}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3aeea25780768830c4c37c3daa7e6e7f4379ed)
![{\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=[f_{1}({\boldsymbol {x}}),f_{2}({\boldsymbol {x}}),\cdots ,f_{m}({\boldsymbol {x}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e98845cc3035bde51b147f8a10fe1ec21ef3c4d8)
![{\displaystyle {\frac {\partial [c_{1}f({\boldsymbol {A}})+c_{2}g({\boldsymbol {A}})]}{\partial {\boldsymbol {A}}}}=c_{1}{\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}+c_{2}{\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8511157a6f2f1874a8cf0eb8357a50711623874d)
![{\displaystyle {\frac {\partial f({\boldsymbol {A}})/g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}={\frac {1}{g({\boldsymbol {A}})^{2}}}\left[g({\boldsymbol {A}}){\frac {\partial f({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}-f({\boldsymbol {A}}){\frac {\partial g({\boldsymbol {A}})}{\partial {\boldsymbol {A}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f83f4e2888fde078ab3c374ce4586e8fa35c181)
