HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 472

Chapter 15 Vector Analysis Applications In this Chapter we present a number of functions from the CALC menu that apply to the analysis of scalar and vector fields. The CALC menu was presented in detail in Chapter 13. In particular, in the DERIV&INTEG menu we identified a number of functions that have applications in vector analysis, namely, CURL, DIV, HESS, LAPL. For the exercises in this Chapter, change your angle measure to radians. Definitions A function defined in a region of space such as φ(x,y,z) is known as a scalar field, examples are temperature, density, and voltage near a charge. If the function is defined by a vector, i.e., F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, it is referred to as a vector field. The following operator, referred to as the 'del' or 'nabla' operator, is a vector- based operator that can be applied to a scalar or vector function: ∇[ ] = i j k ⋅ ∂ [ ] ∂x ∂y ∂z When this operator is applied to a scalar function we can obtain the gradient of the function, and when applied to a vector function we can obtain the divergence and the curl of that function. A combination of gradient and divergence produces another operator, called the Laplacian of a scalar function. These operations are presented next. Gradient and directional derivative The gradient of a scalar function φ(x,y,z) is a vector function defined by grad i j k x ∂y ∂z The dot product of the gradient of a function with a given unit vector represents the rate of change of the function along that particular vector. This rate of change is called the directional derivative of the function, Duφ(x,y,z) = u•∇φ. Page 15-1