Vector moments
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The gradient of a scalar field
Consider a two- dimensional scalar field [Equation]We will define a vector field called the gradient of the scalar field [Equation] by [Equation] Example If [Equation] find grad [Equation] Evaluate grad [Equation] at [Equation] Solution [Equation] We will now show that the direction of grad[Equation] at a point is perpendicular to contour curve passing through that point - that is, it points in the direction of the normal to that contour [Diagram] Proof We have[Equation] as the equation of a contour curve [Diagram] Let [Equation] be a parameterisation of this contour curve Then a vector tangent to this contour curve will be [Equation] [Diagram] Along this curve [Equation] where is a constant. Hence, differentiating with respect to t, [Equation] However, since [Equation] is a function of [Equation] and[Equation] and these are regarded as functions of [Equation], we can apply the chain rule to differentiate [Equation] [Equation] But here[Equation] is the partial denvative of with respect to [Equation goes here - download the original to see it.], and likewise [Equation] is the partial denvative of with respect to y. So, [Equation] Since [Equation], this means [Equation] Now the expression [Equation] is the scalar (dot) product of the two vectors[Equation]. That is [Equation] [Equation] Hence the dot product of these two vectors is zero. [Equation] Hence the vector [Equation]is perpendicular to the vectors. Since the vector [Equation] is tangent to the contour curve, the vector [Equation] is normal to it [Diagram] In three-dimensional space a scalar field is represented by the field function [Equation]. Its gradient is [Equation] Example The gravitational potential at a point [Equation] of a gravitational field is given by [Diagram] where is a constant. For example, the gravitational field surrounding the sun [Diagram] The gravitational potential is inversely proportional to the distance of the point from the centre of the sun [Diagram] Find and show that this points in the direction of the centre of the gravitational field. [Equation The Vector Operator [Equation] [Equation] [Equation]
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Contents of
Vector moments
1 Vector calculus - Scalar Field
2 Contour curves
3 Vector Field
4 Vector field lines
5 Differentiation of scalar and vector products.
6 The gradient of a scalar field
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