Vector Field Function
1. Gradient of a scalar field function
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
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