Two dimensionless unit vectors and lie in a plane at an angle of to each other

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

Two dimensionless unit vectors a1\mathbf{a}_1 and a2\mathbf{a}_2 lie in a plane at an angle of 60∘60^{\circ} to each other, with a1\mathbf{a}_1 along the xx-axis. A scalar potential VV has directional derivative ∇V⋅a1=3 V/m\nabla V\cdot\mathbf{a}_1=3\,\mathrm{V/m} and ∇V⋅a2=−1 V/m\nabla V\cdot\mathbf{a}_2=-1\,\mathrm{V/m}. ((a)) Write down the metric gij=ai⋅ajg_{ij}=\mathbf{a}_i\cdot\mathbf{a}_j of the basis {a1,a2}\{\mathbf{a}_1,\mathbf{a}_2\} and invert it. ((b)) Treating ω(v)=∇V⋅v\omega(\mathbf{v})=\nabla V\cdot\mathbf{v} as a linear functional with components ω1,ω2\omega_1,\omega_2 equal to the two given directional derivatives, use ui=gijωju^i=g^{ij}\omega_j to find ∇V\nabla V in Cartesian (x,y)(x,y) components. ((c)) State the magnitude of ∇V\nabla V.

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