Scalar Function:

Generally, What Is Scalar Function?

The Answer Is that a scalar function may be defined as A function of one or more variables whose range is one-dimensional, as compared to a vector function, whose range is three-dimensional (or, in general, -dimensional). Scalar Field

When We Talk about Scalar Field, We Are Talking about the Scalar Function Being Applied to a Space (More like Euclenoid Space etc) or, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space (or space-time) Gradient

Gradient of Scalar Field Function (E.g. Pressure, Temperature etc) will be the Vectors Which Would Eventually Point towards the Direction of Maximum Magnitude Increase. Temperature Gradient (Gradient of Scalar function “Temperature”)

A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature gradient is a dimensional quantity expressed in units of degrees (on a particular temperature scale) per unit length. The SI unit is kelvin per meter (K/m). The Application(s):

Weather and climate relevance

Differences in air temperature between different locations are critical in weather forecasting and climate. The absorption of solar light at or near the planetary surface increases the temperature gradient and may result in convection (a major process of cloud formation, often associated with precipitation). Similarly, on a global and annual basis, the dynamics of the atmosphere (and the oceans) can be understood as attempting to reduce the large difference of temperature between the poles and the equator by redistributing masses of warm and cold air (and water). Meteorological fronts are regions where the horizontal temperature gradient may reach relatively high values, as these are boundaries between air masses with rather distinct properties. Clearly, the temperature gradient may change substantially in time, as a result of diurnal or seasonal heating and cooling for instance. Day to day experiences and health issues

Other places where noticeable temperature gradients can be experienced include the entrance (or exits) of air conditioned shops in the summer, or the entrance of caves and other protected or poorly ventilated areas. Rapid changes in temperature (in space or time) may cause discomfort and, in extreme cases, heat or cold stresses.

2.Divergence of a Vector Field Function :

Vector Function

A mathematical function of one or more variables whose range is a set of multidimensional vectors (3 Dimensional). The input of a vector function could be a scalar or a vector. The dimension of the domain is not defined by the dimension of the range.

Vector Field

A vector field is assigning of a vector to each point in of Euclidean space. A vector field in the plane, for example, can be seen as group of arrows having some magnitude and direction set to a definite point in that space. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force.

Divergence

The divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. It is a local measure of its "outgoingness"—the extent to which there is more exiting an infinitesimal region of space than entering it. Practical Applications

Divergence represents the flux of a quantity through a differential volume and has significance in fluid mechanics in particular. Not only can it tell you whether a point is a...