3- The plane X- y + 3z = 6 marks the boundary betwee two regions. lat Region I defined by x-y 13276 with M = 340, and Region 2 is defined by x-y rsz < 6 with μ = 6 Mo. Assume that H₁ = 200 + lody - loaz A/m, Find anız a- b - Hin and H₂n C-H₂ and B₂ B₂ d - The angles. "Hi and He make with the boundary magnetic energy densities in the two regions. e- the the

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Gradient of a Scalar Field

Consider a room or a place where the temperature is denoted by a scalar field T, such that at each point (x, y, z), the temperature is given as T(x, y, z). It is independent of time, in which the temperature rises most quickly. The gradient of T will show the direction at that point by moving away from (x, y, z). How far will the rise in temperature in that direction be determined by the gradient magnitude? Consider a surface that has a height above sea level at point (x, y) which is given by H(x, y). At that point, pointing in the direction of the steepest slope or grade is the gradient of H. The magnitude of the gradient of the vector gives the steepness of the slope at that point. How a scalar field changes in another direction can be measured by the gradient by taking a dot product rather than just the direction of the greatest change?