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Chapter 10 Vector Calculus
10.1 Differentiation of vector
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Chapter 10 Vector Calculus
For two-dimensional plane polar coordinates
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Chapter 10 Vector Calculus
Differentiation of composite vector expressions
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Chapter 10 Vector Calculus
10.2 Integration of vector
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Chapter 10 Vector Calculus
10.3 Space curves Ex: A curve lying in the xy-plane is given by y=y(x), z=0. Evaluate the arc length along the curve between x=a and x=b.
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Frenet-Serret formula:
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
10.4 Vector functions of several arguments
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Chapter 10 Vector Calculus
Surface
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Ex: Find the element of area on the surface of a sphere of radius a, and hence calculate the total surface area of the sphere.
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Chapter 10 Vector Calculus
Vector operator Gradient of a scalar field
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Chapter 10 Vector Calculus
Ex: For a function at a point (1,2-1), find its rate of change with distance in the direction At the same point, what is the greatest possible rate of change with distance and in which direction does it occur?
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Ex: Find the expression for the equations of the tangent plane and the line normal to the surface at the point P with the coordinates Use the results to find the equations of the tangent plane and the line normal to the surface of the sphere at the point
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Chapter 10 Vector Calculus
Divergence of a vector field Scalar differential operator
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Chapter 10 Vector Calculus
Curl of a vector
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Useful formulas:
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Chapter 10 Vector Calculus
10.8 Vector operator formula Useful special cases:
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Combinations of grad, div, and curl
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
10.9 Cylindrical and spherical polar coordinates A. Cylindrical polar coordinates
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
B. Spherical polar coordinates
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
General curvilinear coordinate
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
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Chapter 10 Vector Calculus
Gradient Divergence Proof:
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Chapter 10 Vector Calculus
Laplacian
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Chapter 10 Vector Calculus
Curl Proof:
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