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Unformatted text preview: Vector Calculus in the Plane John E. Gilbert, Heather Van Ligten, and Benni Goetz Since this is a calculus class, its time to introduce the derivative of a vector-valued function. So let r ( t ) = x ( t ) i + y ( t ) j = ( x ( t ) , y ( t ) ) be a vector-valued function whose components x ( t ) , y ( t ) well assume are differentiable on some interval a < t < b . The definition of the derivative r ( t ) is just the same as for a real-valued function - whats important is interpreting the limit of the Newtonian Quotient r ( t ) = lim h r ( t + h ) r ( t ) h in terms of vectors! By the Triangle Law, r ( t + h ) r ( t ) is the dis- placement vector PQ ; its direction is the secant line shown in pink through P and Q . So the Newtonian Quotient can be identified as a vector, the scalar multiple 1 h parenleftBig r ( t + h ) r ( t ) parenrightBig . Its shown in black and has the direction of the secant line through P and Q . Now let h . The secant lines approach the tangent line at P to the graph of r shown in red, while the black vectors approach the tangent vector to the graph of r at P . r ( t ) r ( t + h ) r ( t ) P Q Thus the vector r ( t ) can be identified with the tangent vector to the graph of r at P ; it gives the rate of change of r ( t ) at P . To compute r ( t ) in coordinates: r ( t ) = lim h r ( t + h ) r ( t ) h = lim h ( x ( t + h ) i + y ( t + h ) j ) ( x ( t ) i + y ( t ) j ) h = lim h parenleftBigg x ( t + h ) x ( t ) h parenrightBigg i + lim h parenleftBigg y ( t + h ) y ( t ) h parenrightBigg j , since we know how to subtract vectors and take scalar multiples. Thus differentiation is donesince we know how to subtract vectors and take scalar multiples....
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