Review of Partial Derivatives - Bruce E. Shapiro

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Vector Products There are two types of products between vectors, one of which produces a vector and the other produces a scalar • The dot product v r ⋅w r ⎯→⎯ scalar • The cross product v r × w r ⎯→⎯ vector The Dot Product is defined geometrically r r v⋅ w = v wcosθ where q is the angle between the two vectors as shown in the figure. Algebraically, if r r r r r r r r v = iv1+ jv2 + kv3 and w = iw1+ jw2 + kw3 r r r r Then v⋅ w = w⋅ v = v1w1+ v2w2 + v3w3 Example. Suppose that r r r r u = 3i + 4 j + 5 k and r r r r v = 7i + 8j + 9k Then u r ⋅ v r = ( 3)( 7) + ( 4)( 8) + ( 5)( 9) = 21 + 32 + 45 = 98 Properties of the dot product r r r r 1. v⋅ w = w⋅v r r r r r r 2. v⋅ ( aw) = ( av) ⋅ w = a( v⋅w) 3. ( v r + u r ) ⋅ w r = v r ⋅ w r + u r ⋅w r r 4. v and ware r perpendicular only if v r ⋅ w r = 0. The Cross Product The cross product is a product between vectors that results in a vector. It is defined as a vector with the following properties: • Its length is equal to v r × w r = v r w r sinθ • direction is perpendicular to the plane that contains v and w • Its orientation (up vs. down) is according to the right hand rule Right-Hand Rule: Place u r and v r so that their tails coincide and curl the fingers of your right hand from through the angle from u r to v r . Your thumb is pointing in the direction of u r × v r The cross product gives the area of the parallelogram formed by the two vectors: θ θ w|sinθ v| Page 4
We can also calculate the cross product algebraically from the components of the individual vectors if we use determinants. r r r i j k r r r r r v w v v v i v 2 v 3 j v 1 v 3 k v 1 v 3 × = 1 2 3 = − + w2 w3 w1 w3 w1 w3 w1 w2 w3 r r r = i( v w −v w )− j( v w −v w )+ k( v w −v w ) 2 3 3 2 1 3 3 1 1 2 2 1 Determinant of a Matrix ⎛ det a b ⎞ a b ⎜ ⎟ ≡ ≡ ad −bc ⎝ c d⎠ c d ⎛ a b c⎞ a b c det⎜d e f⎟ d e f a e f b d f c d e ⎜ ⎟ = = h i − g i + g h ⎝ g h i⎠ g h i = a( ei − fh) −b( di − fg) + c( dh −eg) Properties of the Cross Product (1) w r × v r = − v r × w r r r r r r r (2) ( av)× w = a( v × w)= v ×( aw) (3) u r × ( v r + w r )= u r × v r + u r × w r (4) u r × v r = 0 if and only if the vectors are parallel (assuming that r r r uv , ≠ 0) (5) i r × r j = k r , r j × k r = i r , k r × i r = r j (cyclic cross products) (6) r j × i r = − k r , k r × r j = − i r , i r × k = − r j (acyclic cross products) (7) v r × v r = 0 Vector Operators: Gradient, Divergence and Curl The gradient of a function of three variables is the vector r f f grad f ≡∇f( x, y, z) ≡i ∂ j k f ∂ x + r ∂ ∂ y + r ∂ ∂z We can think of the gradient symbol ∇ as a operator that behaves exactly like a vector with the exception of the fact that it operates on the single item immediately to its right. In this sense, we can think of ∇ as vector version of ∂ ∂x . Specifically, ∇ represents the “vector” r r ∂ r r ∇= ∂ + ∂ ∂ + ∂ i j k x y ∂ z Page 5
Page 1 and 2: Review of Partial Derivatives Defin
Page 3: Chain Rule If zt () = f( xt (), yt
Page 7 and 8: The Curl Operator is defined as “
Page 9: Therefore rˆ = cosθi ˆ + sinθj

Review of Partial Derivatives - Bruce E. Shapiro

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