H + ) {\displaystyle \nabla \times (c\mathbf {G} )=c(\nabla \times \mathbf {G} )} + ( ) G 2 ^ {\displaystyle \nabla (\mathbf {F} \cdot \mathbf {G} )=(i,{\frac {\partial }{\partial x_{i}}}(\mathbf {F} \cdot \mathbf {G} ))} {\displaystyle =(i,(\nabla ^{2}f)G_{i}+2(\nabla f)\cdot (\nabla G_{i})+f(\nabla ^{2}G_{i}))} ( ∇ + ) F {\displaystyle =(i,v((\mathbf {V} \cdot \nabla )F_{i}))} ∂ ρ F + ( ) ( ∇ ) F G ( r F i Every vector space has a unique additive identity. ( j H V ) cos 1 ⊥ i ∂ ) ∂ ∇ G ∂ i i 2 F x We have no intristic reason to believe these identities are true, however the proofs of which can be tedious. i ∂ ) , x i ) × ∂ ϕ ^ ) ∂ i + ( ∂ i ) ) V {\displaystyle f} ∑ C) B - (A . ϕ n ) i θ 1 i g r ⁡ + x ∇ i f V ) x ∂ {\displaystyle =v_{r}\mathbf {0} +{\frac {v_{\theta }}{r}}(-{\hat {\mathbf {r} }})+{\frac {v_{\phi }}{r\sin \theta }}(\cos \theta {\hat {\mathbf {\phi } }})} ^ i + i i + ) ∇ ∇ ∂ + ( ⁡ {\displaystyle \nabla \times (\nabla f)=\nabla \times (i,{\frac {\partial f}{\partial x_{i}}})} ) ∂ x ⋅ ∂ ( i ) ( x ) ) ^ ( G + ) ( + 2 , {\displaystyle =(i,F_{i}({\frac {\partial G_{i+1}}{\partial x_{i+1}}}+{\frac {\partial G_{i+2}}{\partial x_{i+2}}})-({\frac {\partial F_{i+1}}{\partial x_{i+1}}}+{\frac {\partial F_{i+2}}{\partial x_{i+2}}})G_{i}-(F_{i+1}{\frac {\partial G_{i}}{\partial x_{i+1}}}+F_{i+2}{\frac {\partial G_{i}}{\partial x_{i+2}}})+({\frac {\partial F_{i}}{\partial x_{i+1}}}G_{i+1}+{\frac {\partial F_{i}}{\partial x_{i+2}}}G_{i+2}))}, = … x ( j ρ 2 ∂ {\displaystyle =\sum _{i}({\frac {\partial f}{\partial x_{i}}}G_{i})+f\sum _{i}{\frac {\partial G_{i}}{\partial x_{i}}}} i C 2 ∇ + . G ∂ + The main thing to appreciate it that the operators behave both as vectors and as differential operators, so that the usual rules of taking the derivative of, say, a product must be observed. + × + G ∂ f 1 Vector Identities As an example, we will derive the simple vector identities using . + ∂ x H ∑ {\displaystyle \mathbf {F} } ) ) ��I�y��/��TFe��u-�u�Y^�O�*ߔ���ҷUQ�L@c]m\��.���j%Px�����D�8{�Qxu��Ez= ���$�YW���"!�6O��ۺ��� �-i�+'��+�G�"o_�M. ( x {\displaystyle f} W G i 2 f + ( ) ( ∂ 1 g ϕ x {\displaystyle \nabla \times (\mathbf {F} \times \mathbf {G} )=((\nabla \cdot \mathbf {G} )\mathbf {F} +(\mathbf {G} \cdot \nabla )\mathbf {F} )-((\nabla \cdot \mathbf {F} )\mathbf {G} +(\mathbf {F} \cdot \nabla )\mathbf {G} )}, ∇ 1 ϕ H G 1 = + {\displaystyle =(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)}, ∇ ) ^ {\displaystyle =(i,({\frac {\partial ^{2}F_{i+1}}{\partial x_{i}\partial x_{i+1}}}+{\frac {\partial ^{2}F_{i+2}}{\partial x_{i}\partial x_{i+2}}})-({\frac {\partial ^{2}F_{i}}{\partial x_{i+1}^{2}}}+{\frac {\partial ^{2}F_{i}}{\partial x_{i+2}^{2}}}))} G ∇ 1 | i f F ) ∂ 1 x ⋅ 2 ) {\displaystyle =\sum _{i}(V_{i}{\frac {\partial f}{\partial x_{i}}})+\sum _{i}(W_{i}{\frac {\partial f}{\partial x_{i}}})} ⋅ ) ( θ ( 1 Then 0 ′= 0+0 = 0, where the first equality holds since 0 is an identity and the second equality holds since 0′ is an identity. + ) + 1 r x {\displaystyle (\mathbf {V} \cdot \nabla )(\mathbf {F} \times \mathbf {G} )=((\mathbf {V} \cdot \nabla )\mathbf {F} )\times \mathbf {G} +\mathbf {F} \times ((\mathbf {V} \cdot \nabla )\mathbf {G} )}, ( ϕ + , f 2 2 i ) ∇ g f G ⋅ ) G i i 2 V = v ∂ i)×( . v ∂ 2 ) G θ ( j ) 2 i ( f ( ∂ × ∂ f i + i v + + ( g ( ) r ) ( i ∂ ∂ v . ) ( ∂ n ϕ + ( = + F First, lets do the scalar triple product of vectors. + z + ∂ + ∇ + 2 ∂ + + ∂ ^ + ∇ V Every vector space has a unique additive identity. x ∇ i θ x ( θ ∂ ( 2 G ) f i ( = i V G However, in other coordinate systems like cylindrical coordinates or spherical coordinates, the basis vectors can change with respect to position. 1 2 + ^ ∇ ∂ ( = ) ∂ ∇ ) i : + + 2 {\displaystyle ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}\mathbf {0} )+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}(\sin \theta (-{\hat {\mathbf {r} }}))+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial }{\partial \phi }}(\cos \theta {\hat {\mathbf {\phi } }})} 2 ) ( ) ) x i ⋅ V , ) ∂ ∂ G and i x ∂ F {\displaystyle \mathbf {k} } i r ^ {\displaystyle \nabla \times (f\mathbf {G} )=(\nabla f)\times \mathbf {G} +f(\nabla \times \mathbf {G} )} + ( ϕ i i 2 , i 1 + {\displaystyle =(i,({\frac {\partial ^{2}F_{i}}{\partial x_{i}\partial x_{i}}}+{\frac {\partial ^{2}F_{i+1}}{\partial x_{i}\partial x_{i+1}}}+{\frac {\partial ^{2}F_{i+2}}{\partial x_{i}\partial x_{i+2}}})-({\frac {\partial ^{2}F_{i}}{\partial x_{i}^{2}}}+{\frac {\partial ^{2}F_{i}}{\partial x_{i+1}^{2}}}+{\frac {\partial ^{2}F_{i}}{\partial x_{i+2}^{2}}}))} i = 2 ∂ F ∂ r i + ( + ∂ First Edition (version 1.0) published online on 08 May 2009 This file shall be a good reference to vector identities and their proofs. G ⋅ … 2 θ + A list of these vector identities is provided and for each one also is provided a proof … ) f ∇ i {\displaystyle \nabla ^{2}\mathbf {F} } ∂ ) I'm just going to assume-- let's say I have vector a. 2 i , ϕ 1 2 ) ∑ 2 − F ) {\displaystyle \mathbf {G} } ∂ j i i ∂ G = ∂ = {\displaystyle \mathbf {V} _{\perp }=v_{\rho }{\hat {\mathbf {\rho } }}+v_{z}{\hat {\mathbf {z} }}}. ^ i {\displaystyle g} ⋅ ) ∑ ( i ∂ = + ∂ G {\displaystyle \mathbf {F} } {\displaystyle f} i Proof. i g F θ When ∂ = + ( i y ∂ ∑ , ∇ = ∂ V r ( ) {\displaystyle =(i,(({\frac {\partial F_{i}}{\partial x_{i+1}}}G_{i+1}+F_{i}{\frac {\partial G_{i+1}}{\partial x_{i+1}}})-({\frac {\partial F_{i+1}}{\partial x_{i+1}}}G_{i}+F_{i+1}{\frac {\partial G_{i}}{\partial x_{i+1}}}))-(({\frac {\partial F_{i+2}}{\partial x_{i+2}}}G_{i}+F_{i+2}{\frac {\partial G_{i}}{\partial x_{i+2}}})-({\frac {\partial F_{i}}{\partial x_{i+2}}}G_{i+2}+F_{i}{\frac {\partial G_{i+2}}{\partial x_{i+2}}})))} ) G F ) n {\displaystyle \nabla \cdot (\mathbf {F} +\mathbf {G} )=\sum _{i}({\frac {\partial }{\partial x_{i}}}(F_{i}+G_{i}))} + θ {\displaystyle =(\mathbf {V} \cdot \nabla )f+(\mathbf {V} \cdot \nabla )g}, ( v r ∑ ( ∂ ( C = , ( F V ) sin V ) ( 2 + 0 ( F i = i ∂ ∇ ( θ ∑ ( ( ( ∂ x ∂ ) θ ∂ + {\displaystyle \nabla (f+g)=(\nabla f)+(\nabla g)} i ( i ∂ + i Here a⃗×(b⃗×c⃗)\vec a \times (\vec b \times \vec c)a×(b×c) is coplanar with the vectors b⃗andc⃗\vec b\ and\ \vec cbandc and perpendicular to a⃗\vec aa. 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