( ∂ {\displaystyle =(\mathbf {V} \cdot \nabla )\mathbf {F} +(\mathbf {V} \cdot \nabla )\mathbf {G} }, Given vector field ) , i ) ( f + F , j If e = ∂ ( ) = n , ∂ i i cos θ ⋅ ⋅ 1 2 ) 2 = 2 + ∂ ⁡ i ∂ ∂ ϕ 1 , the Laplacian is generally written as: When the Laplacian is equal to 0, the function is called a Harmonic Function. i ∂ f = div ⋅ r v 2 . 2 i 1 {\displaystyle (\mathbf {V} \cdot \nabla )f} {\displaystyle =\sum _{i}V_{i}({\frac {\partial f}{\partial x_{i}}}g+f{\frac {\partial g}{\partial x_{i}}})} F A 2 f j 1 ∇ i = ∑ V ^ {\displaystyle =((\nabla \cdot \mathbf {G} )\mathbf {F} +(\mathbf {G} \cdot \nabla )\mathbf {F} )-((\nabla \cdot \mathbf {F} )\mathbf {G} +(\mathbf {F} \cdot \nabla )\mathbf {G} )}, The following identity is a very important property of vector fields which are the gradient of a scalar field. ( i + G ) x 0 ∂ + ^ f i operations are understood not to act on the v i ϕ i ( is the directional derivative in the direction of i ( + ( F ⋅ i ) 1 {\displaystyle =(i,(\mathbf {V} \cdot \nabla )F_{i}+(\mathbf {V} \cdot \nabla )G_{i})} ( f θ x ( r ) i input function i ( ( g + 1 F i ∂ ( i = 2 ) 1 i + + = + v g ( 2 i V f F ∂ + i ρ g We have the following generalizations of the product rule in single variable calculus. ⁡ ∂ ( ) ) 2 i θ ) ^ + ∂ F i ⋅ i ∂ ^ and F ( F x F ) ^ ∂ + ⁡ ⋅ , ) i G + − , ( ) + ) ( ∇ ( + i ⁡ F 2 ) 2 ( ∑ i i x F ∂ ∑ V 1 Vector calculus is also known as vector analysis which deals with the differentiation and the integration of the vector field in the three-dimensional Euclidean space. It means that you can think about the double integral being related to the line integral. ∂ x ∇ 1 i ^ r i ( ) f ( 2 ) ) − i ) G ) c ⋅ = ( ) θ 2 − = ) cos r + +

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