1. What is the relative increase or decrease in conductivity of a conductor if the area is reduced by 30% and the length is increased by 40%? The resistivity is fixed. 2. (a) How much energy (in kilowatthours) is required to light a 60-W bulb continuously for 1 year (365 days)? (b) How long can a 205-W television set be on before using more than 4 kWh of energy?

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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?