Define the following concepts in a few short sentences What

Define the following concepts in a few short sentences. What is the physical meanings of each concept? Gradient of a scalar field CURL of a vector filed Divergence of a scalar field

Solution

a)The gradient (or gradient vector field) of a scalarfunction f(x1, x2, x3, ..., xn) is denoted f or f {\\displaystyle {\\vec {\ abla }}f} where (the nabla symbol) denotes the vector differential operator, del. The notation \"grad(f)\" is also commonly used for the gradient.

b)In vector calculus, the curl is a vector operatorthat describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by avector. The attributes of this vector (length and direction) characterize the rotation at that point.The alternative terminology rotor or rotationaland alternative notations rot F and × F are often used (the former especially in many European countries, the latter, using the deloperator and the cross product, is more used in other countries) for curl and curl F.

c)In vector calculus, divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field\'s source at each point. More technically, the divergence represents the volume density of the outwardflux of a vector field from an infinitesimal volume around a given point.

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the correspondingbasis of unit vectors. The divergence of acontinuously differentiable vector fieldF = Ui + Vj + Wk is defined as the scalar-valued function:

{\\displaystyle \\operatorname {div} \\,\\mathbf {F} =\ abla \\cdot \\mathbf {F} =\\left({\\frac {\\partial }{\\partial x}},{\\frac {\\partial }{\\partial y}},{\\frac {\\partial }{\\partial z}}\ ight)\\cdot (U,V,W)={\\frac {\\partial U}{\\partial x}}+{\\frac {\\partial V}{\\partial y}}+{\\frac {\\partial W}{\\partial z}}.} [{\\displaystyle \\operatorname {div} \\,\\mathbf {F} =\ abla \\cdot \\mathbf {F} =\\left({\\frac {\\partial }{\\partial x}},{\\frac {\\partial }{\\partial y}},{\\frac {\\partial }{\\partial z}}\ ight)\\cdot (U,V,W)={\\frac {\\partial U}{\\partial x}}+{\\frac {\\partial V}{\\partial y}}+{\\frac {\\partial W}{\\partial z}}.}]