Gradient, divergence, curl, and Laplacian operators from "summary" of Differential Calculus by S Balachandra Rao
The operators gradient, divergence, curl, and Laplacian play a crucial role in vector calculus. Let us delve deeper into each of these operators to understand their significance and applications. The gradient operator, denoted by ∇, is used to find the rate of change of a scalar field in a given direction. It is essentially a vector that points in the direction of the steepest increase of the scalar field. Mathematically, the gradient of a scalar function f(x, y, z) is given by ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k, where i, j, and k are the unit vectors along the x, y, and z axes respectively. Moving on to the divergence operator, denoted by ∇·, it is used to determine the extent to which a vector field flows outward or inward from a given point. In other words, it measures the rate at which the field's intensity changes at a particular point. Mathematically, the divergence of a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is given by ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z. The curl operator, denoted by ∇ x, is used to measure the rotation or circulation of a vector field around a point. It gives the direction and strength of the vortex at that point. Mathematically, the curl of a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is given by ∇ x F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k. Lastly, the Laplacian operator, denoted by ∇², is used to measure the divergence of the gradient of a scalar field. It is essentially a scalar operator that gives the scalar value representing the amount of divergence at a given point. Mathematically, the Laplacian of a scalar function f(x, y
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