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Divergence of a vector Field. Let V(x,y,z) be a differentiable Vector function where x,y,z are Cartesian coordinates in space. Let V1, V2, V3 be the components of V Therefore Div V = (δV1/δx) + (δV2/δy) + (δV3/δz) This is the divergence of V or the divergence of the vector field defined by V Divergence of V is ∇.V = (δ/δx)i + (δ/δy)j + (δ/δz)k . (V1i + V2yj + V3k) = (δV1/δx) + (δV2/δy) + (δV3/δz) Note that (δ/δx)V1 in dot product means partial derivative (δV1/δx) This is just a convenient notation and nothing more. ∇.V means the scalar div V Where as ∇f means the vector grad f Eg V = 4xyi + 5zyj – xz2k Div V = 4y + 5z – 2xz If f (x,y,z) is a twice differentiable scalar function, grad f = (δf/δx)i + (δf/δy)j + (δf/δz)k since div V = (δV1/δx)i + (δV2/δy)j + (δV3/δz)k then div V (grad f) = (δ2f/δx2)i + (δ2f/δy2)j + (δ2f/δz2)k the expression on the right is the Laplacian of f Hence div V (grad f) = (δ2f/δx2)i + (δ2f/δy2)j + (δ2f/δz2)k = ∇2f CURL OF A VECTOR FIELD LET x,y,z be right – handed Cartesian Coordinates in space, and let V(x,y,z) = V1i + V2j + V3k Be or is a twice differentiable vector function The function Curl V = ∇ X V = | i j k | | (δ/δx) (δ/δy) (δ/δz) | | V1 V2 V3 | = ((𝜕V3/𝜕y) – (𝜕V2/𝜕z))i + ((𝜕V1/𝜕z) – (𝜕V3/𝜕x))j + ((𝜕V2/𝜕x) – (𝜕V1/𝜕y))k This is called the curl of the vector function V or the Curl of the vector field defined by V Invariance of the Curl The Length and direction of Curl V are independent of the particular choice of Cartesian Coordinate system in space.