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• In Cartesian coordinates the vector operators for gradient, divergence, rotational and Laplacian are of the form ?? = grad ? = i ??/?x + j ??/?y + k ??/?z. ? ? V = div V = ?V_x/?x + ?V_y/?y + ?V_z/?z. ? × V = curl V = i(?V_z/?y - ?V_y/?z) + j(?V_x/?z - ?V_z/?x) + k(?V_y/?x - ?V_x/?y) = | i j k ; ?/?x ?/?y ?/?z ; V_x V_y V_z |. ?²? = ? ? ?? = div grad ? = ?/?x ??/?x + ?/?y ??/?y + ?/?z ??/?z = ?²?/?x² + ?²?/?y² + ?²?/?z² (the Laplacian). determine the expressions for each operator in coordinates a) Cylindrical b) Spherical 

          • In Cartesian coordinates the vector operators for gradient, divergence, rotational and Laplacian are of the form
?? = grad ? = i ??/?x + j ??/?y + k ??/?z.
? ? V = div V = ?V_x/?x + ?V_y/?y + ?V_z/?z.
? × V = curl V
= i(?V_z/?y - ?V_y/?z) + j(?V_x/?z - ?V_z/?x) + k(?V_y/?x - ?V_x/?y)
= | i j k ; ?/?x ?/?y ?/?z ; V_x V_y V_z |.
?²? = ? ? ?? = div grad ? = ?/?x ??/?x + ?/?y ??/?y + ?/?z ??/?z
= ?²?/?x² + ?²?/?y² + ?²?/?z² (the Laplacian).
determine the expressions for each operator in coordinates
a) Cylindrical
b) Spherical
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• In Cartesian coordinates the vector operators for gradient, divergence, rotational and Laplacian are of the form ?? = grad ? = i ??/?x + j ??/?y + k ??/?z. ? ? V = div V = ?Vx/?x + ?Vy/?y + ?Vz/?z. ? × V = curl V = i(?Vz/?y - ?Vy/?z) + j(?Vx/?z - ?Vz/?x) + k(?Vy/?x - ?Vx/?y) = | i j k ; ?/?x ?/?y ?/?z ; Vx Vy Vz |. ?²? = ? ? ?? = div grad ? = ?/?x ??/?x + ?/?y ??/?y + ?/?z ??/?z = ?²?/?x² + ?²?/?y² + ?²?/?z² (the Laplacian). determine the expressions for each operator in coordinates a) Cylindrical b) Spherical

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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In Cartesian coordinates the vector operators for gradient, divergence, rotational and Laplacian are of the form ∇ϕ = grad ϕ = i ∂ϕ/∂x + j ∂ϕ/∂y + k ∂ϕ/∂z. ∇ ⋅ V = div V = ∂V_x/∂x + ∂V_y/∂y + ∂V_z/∂z. ∇ × V = curl V = i(∂V_z/∂y - ∂V_y/∂z) + j(∂V_x/∂z - ∂V_z/∂x) + k(∂V_y/∂x - ∂V_x/∂y). ∇²ϕ = ∇ ⋅ ∇ϕ = div grad ϕ = ∂/∂x ∂ϕ/∂x + ∂/∂y ∂ϕ/∂y + ∂/∂z ∂ϕ/∂z = ∂²ϕ/∂x² + ∂²ϕ/∂y² + ∂²ϕ/∂z² (the Laplacian). determine the expressions for each operator in coordinates a) Cylindrical b) Spherical
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Transcript

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0:00 Hi.
00:01 Now we have to write the expression for gradient, divergence, curl and laplation in the cylindrical and spherical coordinates.
00:12 So first we will be writing in cylindrical coordinates.
00:14 So cylindrical coordinates are of the form r theta z.
00:17 So let er, e theta and ez be the unit vectors in the direction of increasing r theta and z...
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