9. An electric circuit consists of two components as shown in the figure below. ZI www R₁ XL ooooo Z2 www 1 1 1 + Z ZI Z2 R₂ Xc The generalisation of the resistance and reactance to an AC circuit is a complex number, known as impedance. For the electrical circuit pictured above, the impedance of the first component can be expressed by z₁ = R₁ + X₁i and the impedance of the second component can be expressed by z2 = R2 - Xci, where R₁ = 3 Q2, XL = 39, R₂ = 492, and Xc = 49. For electrical circuits connected in parallel, as shown above, the total impendance z of the circuit can be computed using the relationship (a) Compute the total impedance z of the two components. (b) What are the modulus and principal argument of the total impedance?

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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?