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# vctut02s - The University of Sydney School of Mathematics...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 2 MATH2061: Vector Calculus Summer School 2012 1. Find grad f if f ( x, y ) = x 2 cos xy . Solution: grad f = ∇ f = ∂f ∂x i + ∂f ∂y j = (2 x cos xy − x 2 y sin xy ) i − x 3 sin xy j . 2. Find grad φ if φ ( x, y, z ) = 3 x + 4 y − 8 z . Solution: grad φ = ∇ φ = ∂φ ∂x i + ∂φ ∂y j + ∂φ ∂z k = 3 i + 4 j − 8 k . 3. Find ∇ φ if φ ( x, y, z ) = e x y + 3 xyz . Solution: ∇ φ = grad φ = ∂φ ∂x i + ∂φ ∂y j + ∂φ ∂z k = ( e x y + 3 yz ) i + ( e x + 3 xz ) j + 3 xy k . 4. Calculate Curl F if F = (sinh x ) i + (cosh y ) j − xyz k . Solution: ∇× F = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k ∂ ∂x ∂ ∂y ∂ ∂z sinh x cosh y − xyz vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = − xz i + yz j . 5. Let S be the surface defined by z = xy ....
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