Math 335-001
Math 335-001 * Preparation list for the midterm
Midterm date: October 26, 2005, 8:30am, KUPF 107
1. Vector algebra
• Dot product is a scalar (the component of one vector along another vector…)
• Cross products is a vector; its magnitude is the area of a parallelogram
• Equations for planes (n ∙ r = const) and lines (a × r = n)
• Triple scalar product: a ∙ ( b × c) = (a × b) ∙ c = (c × a) ∙ b = - (a × c) ∙ b=…
• Triple vector product: a × ( b × c) = b (a ∙ c) – c (a ∙ b)
o Use triple products to simplify vector expressions
2. Suffix notation:
o Dummy indices appear twice and represent sums; free indices should match
• Dot product using suffix notation ( a ∙ b = am b m)
• Cross product in suffix notation - the alternating tensor: (a × b) i = ε i m n am b n
• The Kronecker delta (easy to eliminate when simplifying: δ k p X p …= X k … )
• Product of two alternating tensors with a common dummy index as a combination of Kronecker deltas (ε k l m ε m p q= δ k p δ l q – δ k q δ l p)
o εi m n = ε m n i = εn i m = - ε i n m = - ε n m i = - εm i n; δ k p = δ p k
o ε i m n S m n= 0 if S m n = S n m , for instance ε i m n [pic]m[pic]n= 0, ε i m n u m u n= 0, etc.
3. Partial differentiation of vector and scalar fields
• Sketching vector and scalar fields; isosurfaces and isocurves (solve f(x,y,z)=const)
• Gradient of a scalar field (normal to isosurface or isocurve; directional derivative; linear approximation: f(r + dr) - f(r) = [pic]f(r) ∙ dr )
• Divergence of a vector field is a scalar field; curl of a vector field is a vector field
• Second-order partial derivatives (e.g. [pic]×([pic]×u)=[pic]([pic]∙u) -[pic]2u ; [pic]∙[pic]×w=0)
• Product rules for partial derivatives (e.g. [pic] ∙ ( u × v) = ([pic]× u) ∙ v - ([pic] × v) ∙ u ) Use suffix notation for partial differentiation - in suffix notation partial differentiation behaves like ordinary differentiation - usual product rule applies: e.g. [pic]i(g f)= g[pic]if + f[pic]ig
4. Integration:
• Line intergrals and its relation to the curl of a vector field; curve parametrization
• Line integrals and conservative vector fields [pic]
• Surface integrals and its relation to divergence; integrals over curved surfaces:[pic]
• Volume integrals: calculating volume, mass, etc.
• The divergence theorem connects certain surface and volume integrals [pic]
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