MAP2
MATH ASSISTANCE PAGE(MAP)--GRADIENT What is the gradient?The word "gradient" makes us think of the slope of a hill or an embankment and how steepit is or how rapidly height changes with distance. In other words, it describes the rate at which the height of the surface of the land above some reference point varies as we movehorizontally. The height may be represented as a scalar function H(x,y), so that ?H/?x? will represent the rate of elevation increase or decrease as we move in the i-direction.Similarly, ?H/?y represents rate of change along the y-coordinate.The mathematical meaning of the gradient is similar. From the gradient of some quantity(e.g. T=temperature), we can find the rate of change of temperature as we move through space in any direction--either up or down, forward or backward, left or right or some combination of these. You know that temperature will usually vary as you move from one place to the next. To help understandthis, think about some real examples. As you move from indoors to out(i.e. move through space and throughthe door), the temperature will become higher or lower. We could describe this as a mahematical functionT(x,y,z) of position in a rectangular or some other coordinate system. The function T(x,y,z) could be found experimentally by taking a thermometer and measuring the temperature for each value of x, y and z. We may also have a mathematical expression for T(x,y,z). The gradient of the temperature is a spatial derivativethat tells us how fast the temperature changes as we move about. Inside a room, the temperature difference from one spot to the next may be small so the temperature gradient is small. As we go through the door to the outsidehowever, we mgiht expect a large change and a large gradient. Hence, the temperature difference we feel will depend on where we are and what direction we go. As a result, the GRADIENT IS A VECTOR! How do we get the gradient of the temperature? It may be found from the partial derivatives(See MAP1): grad T = ?T/?x i + ?T/?x j +?T/?z k where "grad T" is often represented with the "nabla" symbol, an upside down triangle, in front of T or ?T. The i, jand k are the unit vectors.The vector grad T has a special direction. It is the direction of greatest increase at some location x, y and z.To find the rate of change for some other direction, we take the dot product with a unit vector in that direction.Try this by taking the dot product between grad T above and j(both vectors, usually underlined) to find the rate of changein temperature along the y-direction. Does the result make sense to you? What is the rate of change in the z-direction? Canyou find the rate of change in temperature along the direction described by the unit vector u = .7071 i + .7071 k?What was the gradient of the air temperature in Norman or your home town on January 3, 2003?(See graphic.) ................
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