Vector and Scalar Fields




The physical world is characterized by change -- changes that occur over time and over distance.  The temperature within a room varies with location.  It may be colder near the window on a winter's day and much warmer next to a nearby blazing fire in the same room.  Temperature is an example of a scalar physical quantity; it has a magnitude associated with it, but no directional sense.  Other examples of scalar quantities include pressure, energy, concentration or density.  Each of these quantities may vary with physical location and over time, regardless of how we choose to express the fact.  In mathematical or engineering terms, however, a physical quantity that varies with location can be termed a field and temperature constitutes an example of a scalar field.

It's a common clinical approximation to consider the pressure within a cardiac chamber or blood vessel as being constant at a particular moment in time.  This is a suitable concept for many purposes, but doesn't support the fact that blood is constantly in motion within the heart and circulation.  From a mechanical standpoint blood is, of course, inanimate and responds simply to the physical forces acting upon it.  Blood elements obey the basic Newtonian law of motion, force equals mass multiplied by acceleration.  Every single blood and tissue element of the body is subject to this law, in whole and in part.

While the physical aspects exist apart from the use of mathematics to describe them, fields can be represented mathematically as functions.  We supply the function with the specific location of interest, and the function returns or tells us the value of the physical quantity at that location.  Examine the following function for example:

\( \large T(x,y,z)=4 x + 5 y z + 2\)

Temperature is represented here as T and the coordinates x, y, and z represent position within the coordinate system.  It's typical to list variables that the function depends upon using parentheses as shown above, i.e. \( T(x,y,z)\)    unless the dependency is clear for some other reason.  Dependency on spatial coordinates is common enough that the notation is sometimes abbreviated in different ways.  For example, \( T(\vec{x})\) , \( T(\bar{x})\) , and \( T(\mathbf{x})\)   are other slightly shorter methods to indicate that temperature depends on a vector, the position in space.  The use of x, y, and z for the coordinate designations implies a Cartesian system among engineers; the direction associated with each coordinate doesn't change with location in the space (they are not curved) and each of the axes is perpendicular to the others.

If we are able to represent a physical quantity as a function of space, as suggested above, then we can perform mathematical operations and analyses that greatly enhance our ability to understand the physical world.  The distribution of temperature indicated by the above function implies that temperature varies in a prescribed way, that there is a temperature gradient.  The term gradient is often applied in cardiology when describing pressure variation in the vicinity of a stenotic heart valve (for example).  However cardiologists tend use the term gradient simply to indicate a difference in pressure between two locations and this is not the interpretation employed by other scientists.   The physical units of a gradient are the same physical units of the scalar, but divided by distance.  A temperature gradient, for example, is the rate of change of temperature per centimeter, meter, kilometer, etc.

As you might well imagine, the rate of change of temperature (pressure, concentration, etc.) may depend dramatically on the direction or orientation within the temperature field; i.e. temperature might change more rapidly as we move in the x direction as opposed to the y direction.  Since we already possess the temperature field from the above example, we can express readily the rate of change of temperature in the x direction by determining the derivative of T with respect to x:

\( \large\frac{\partial T}{\partial x}=\large\frac{\partial}{\partial x} (4 x + 5 y z + 2) = 4\)

Here we have determined the partial derivative of the temperature function with respect to x where  \( \Large\frac{\partial}{\partial x}\) is the symbol used to indicate this mathematical operation.  When computing a partial derivative with respect to (wrt) \(x\), anything that doesn't involve \(x\) just looks like a constant, and the derivative of a constant (wrt anything) is \(0\). The partial of \(T\) with respect to \(x\) in the above is equal to 4 indicating that temperature increases by 4 units (Fahrenheit, Celsius) for each distance unit (centimeters, meters) we proceed in the x direction; the derivative of \(5yz+2\) with respect to \(x\) is \(0\).   Apparently, this is the temperature gradient in the x direction regardless of location within the field -- the partial of T with respect to x doesn't vary with position .  However, the temperature also varies in the y and z directions:

\( \large\frac{\partial T}{\partial y}=\large\frac{\partial}{\partial y} (4 x + 5 y z + 2) = 5 z\)

\( \large\frac{\partial T}{\partial z}=\large\frac{\partial}{\partial z} (4 x + 5 y z + 2) = 5 y\)

The temperature gradient in the y direction is 5 z.  Partial differentiation with respect to y in this case did not yield a number (a constant, like 4), but another function, this one depending on location in the z direction.  Similarly, the temperature gradient in the z direction is 5 y in the example; the gradient increases in proportion to location in the y direction.
It is commonplace in mathematical notation to represent the process we have been exploring with a more compact notation:

\(\large \nabla T=\langle \Large\frac{\partial}{\partial x},\large\frac{\partial}{\partial y},\large\frac{\partial}{\partial z}\rangle(4 x + 5 y z + 2) =\langle4,5 z,5 y\rangle\)

The gradient operator,  \( \nabla\) (del), has been introduced and angle brackets \( \langle\rangle\) are being used to represent a vector.  It's understood from this notation that \( \nabla\) means that we will determine the partial derivative in each of the three respective directions of this Cartesian system.  The result of finding the gradient of a scalar field is a vector, a quantity that has both magnitude and direction!  More specifically, it is a vector function -- a function that gives us back a vector when we give it a position, i.e. a particular \( \langle x, y, z\rangle\) .

Let us consider the interpretation of this rather amazing thing.  We now have a new function, the gradient of the temperature ( T) that was derived from the original temperature function.  If we pick a particular location in the temperature field, point \( \langle3, 2, 0\rangle\) for example, the new function gives us a vector back equal to \( \langle4, 5z, 5y\rangle\) or \( \langle4, 0, 10\rangle\) which is a vector that indicates both the magnitude and direction of maximal change in the temperature field at this location, 4 degrees per centimeter in the x direction and 10 degrees per centimeter in the z direction (no change in the y direction).  We have also derived a new kind of field, a vector field.  A vector field associates a vector with each location in the space.   Familiar examples of vectors include velocity, acceleration, and force.  We have just determined that the gradient of a scalar field is also a vector field.

This image depicts the scalar pressure field surrounding a cylinder (on cross section) as flow proceeds from left to right.  There is a pressure value associated with each point in the two-dimensional field, i.e. pressure is a continuous function of position.  Color represents the physical value of the pressure with higher pressure represented by warmer colors.  Contours are shown also to represent lines of equal pressure.  Also superimposed are vectors that represent the gradient of the pressure field (actually the gradient of the negative pressure field, \( \nabla(-P) = -\nabla P\) ).   The gradient of a scalar field is a vector field that associates a vector with each of the points in the field.  It can be appreciated that the vectors represent both the magnitude and direction of change of the scalar field.  Each of the vectors crosses local contour lines perpendicularly and the magnitude of the vector increases as the pressure field changes more abruptly with distance.  In the flow fields that follow, pressure gradient is the only force acting on the fluid.  Consequently the vectors shown also depict the acceleration of the fluid elements.


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