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By Pianigiani G.

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Then the square root of both sides is taken. 2a) Quadratic equations occur frequently in physics and engineering; the formulae for the two roots should be remembered. e. the quadratic is expressed as the product of two linear expressions. It should be noted that there are cases when no real solutions exist. e. the square root can be extracted. It is not hard to see that this corresponds to the function f (x) = ax 2 + bx + c having zeros. e. its graph does not cut the x-axis. We therefore have a criterion which allows us to decide whether a parabola, given by an algebraic expression, lies entirely above the x-axis or below the x-axis; the condition is b 2 − 4ac < 0.

Now F ⊥ = F sin ˛ in magnitude; hence C = rF sin ˛. Fig. 2 Torque as a Vector Physically, torque is a vector quantity since its direction is taken onto account. The following convention is generally accepted. The torque vector C is perpendicular to the plane containing the force F and the radius vector r. The direction of C is that of a screw turned in a way that brings r Fig. 15 32 2 Vector Algebra II: Scalar and Vector Products by the shortest route into the direction of F . This is called the right-hand rule.

In this book we will mainly be concerned with real functions, as opposed to more general functions like complex functions. Note that the concept of a function implies that the y-value is determined unambiguously. During the previous one or two decades the use of the term ‘function’ has changed. In the engineering literature, the term ‘two-valued’ or ‘many-valued function’ is still occasionally used. Strictly speaking, in modern terminology what is meant is not a function but a relationship. g. if y1 = +2 and y2 = −2 then x = 4 in each case.

### A Baire category approach in existence theory of differential equations by Pianigiani G.

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