20. Consider the electric circuit illustrated below. Let I be the current through the inductor and V be the voltage across the capacitor. Then I and V satisfy the linear system: L IP dt (4). R1 I+ L %3D R, AP dt 1 I + %3D CR2 Let R1 = 1N, R2 =N, L = 2 H, and C = ? F. Answer the following questions about the system: (a) Express the system in matrix notation: T' = Pï () (b) Verify that ẽ = is an eigenvector of P. Identify the corresponding eigenvalue A1. (c) Given that 2 = -2 is an eigenvalue of P, identify an eigenvector 2 that corresponds to A2. (d) Find the general solution to the system of differential equations. (e) Show that I(t) → 0 and V (t) → 0 as t →∞, regardless of the initial values of I(0) and V(0).

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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?