Gradient of a scalar field and the calculus of surfaces
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The gradient of a scalar field
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Consider a two- dimensional scalar field [Equation goes here - download the original to see it.]We will define a vector field called the gradient of the scalar field by, [Equation goes here - download the original to see it.] Example If [Equation goes here - download the original to see it.] find grad [Equation goes here - download the original to see it.] Evaluate grad [Equation goes here - download the original to see it.] at [Diagram goes here - download the original to see it.] Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it] We will now show that the direction of grad The gradient of a scalar field 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 goes here - download the original to see it. Proof We have[Equation goes here - download the original to see it.] as the equation of a contour curve. [Diagram goes here - download the original to see it.] Let [Equation goes here - download the original to see it.] be a parameterization of this contour curve. Then a vector tangent to this contour curve will be [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Along this curve [Equation goes here - download the original to see it.] where is a constant. Hence, differentiating with respect to t, [Equation goes here - download the original to see it.] However, since [Equation goes here - download the original to see it.] is a function of and and these are regarded as functions of , we can apply the chain rule to differentiate [Equation goes here - download the original to see it.].[Equation goes here - download the original to see it.] But here[Equation goes here - download the original to see it.] is the partial derivative of[Equation goes here - download the original to see it.] with respect to , and likewise [Equation goes here - download the original to see it.] is the partial derivative of with respect to y. So, [Equation goes here - download the original to see it.] Since [Equation goes here - download the original to see it.] , this means [Equation goes here - download the original to see it.]Now the expression [Equation goes here - download the original to see it.] is the scalar (dot) product of the two vectors [Equation goes here - download the original to see it.]. That is [Equation goes here - download the original to see it.] Hence the dot product of these two vectors is zero. [Equation goes here - download the original to see it.] Hence the vector [Equation goes here - download the original to see it.] is perpendicular to the vector Since the vector [Equation goes here - download the original to see it.] is tangent to the contour curve, the vector [Equation goes here - download the original to see it.] is normal to it [Diagram goes here - download the original to see it.] In three-dimensional space a scalar field is represented by the field function [Equation goes here - download the original to see it.] . Its gradient is [Equation goes here - download the original to see it.]
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Contents of
Gradient of a scalar field and the calculus of surfaces
1 The gradient of a scalar field
2 Example - The gradient of a scalar field
3 The Vector Operator nabla or del
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