Fields and vector calculus The electric potential is a scalar field. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. For example: (a) Suppose we want to model the flow of air around an aeroplane. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. The electric potential is a scalar field. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The electric potential is a scalar field. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. The electric potential is a scalar field. Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). The electric potential is a scalar field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. The electric potential is a scalar field. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. The electric potential is a scalar field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. The electric potential is a scalar field. Vector fields and scalar fields In many applications, we do not consider individual vectors or scalars, but functions that give a vector or scalar at every point. Such functions are called vector fields or scalar fields. For example: (a) Suppose we want to model the flow of air around an aeroplane. The velocity of the air flow at any given point is a vector. These vectors will be different at different points, so they are functions of position (and also of time). Thus, the air velocity is a vector field. Similarly, the pressure and temperature are scalar quantities that depend on position, or in other words, they are scalar fields. (b) The magnetic field inside an electrical machine is a vector that depends on position, or in other words a vector field. The electric potential is a scalar field. Although we will mainly be concerned with scalar and vector fields in three-dimensional space, we will sometimes use two-dimensional examples because they are easier to visualise. , and similarly for any other value of x. 2 I When (x; y) = (0; 0:4) we have u = ((1 − 0:4 )=5; 0) = (0:168; 0) I When (x; y) = (1:2; 0:4) we again have u = (0:168; 0) 2 I When y = 0:8 we have u = ((1 − 0:8 )=5; 0) = (0:072; 0) 2 I When y = −0:8 we have u = ((1 − (−0:8) )=5; 0) = (0:072; 0) The vector field u = ((1 − y 2)=5; 0) y=1 y=−1 , and similarly for any other value of x.