Spherical polar basis and volume element.

<< CliffordBasic`;
$SetSignature={3,0};
Import["https://raw.githubusercontent.com/jkuczm/MathematicaCellsToTeX/master/NoInstall.m"]

Conventional vector algebra only

ClearAll[x,xr, xt, xp,e1,e2,e3, r, theta, phi]
{e1,e2,e3} = IdentityMatrix[3];

x[r_, t_, p_] =  r (e1 Sin[t] Cos[p]+e2 Sin[t] Sin[p]+e3 Cos[t])
xr[r_, theta_, phi_] = D[x[a,theta,phi],a] /. {a -> r, t->theta, p->phi};
xt[r_, theta_, phi_] = D[x[r,t,phi],t] /. {a -> r, t->theta, p->phi};
xp[r_, theta_, phi_] = D[x[r,theta,p],p] /. {a -> r, t->theta, p->phi};

{x[r, θ, ϕ],
xr[r, θ, ϕ],
xt[r, θ, ϕ],
xp[r, θ, ϕ]} // Column

{xr[r, θ, ϕ].xt[r, θ, ϕ],
xr[r, θ, ϕ].xp[r, θ, ϕ],
xp[r, θ, ϕ].xt[r, θ, ϕ],
xr[r, θ, ϕ].xr[r, θ, ϕ],
xp[r, θ, ϕ].xp[r, θ, ϕ],
xt[r, θ, ϕ].xt[r, θ, ϕ]} // FullSimplify

Det[ {xr[r, θ, ϕ],
xt[r, θ, ϕ],
xp[r, θ, ϕ]}] // Simplify

Now, only with GA.

ClearAll[i,j,ej, xg]
i = e[1,2];
j[phi_] = GeometricProduct[e[3,1],Cos[phi] + i Sin[phi]];
ej[t_, p_] = Cos[t] + j[p] Sin[t];
xg[r_, t_, p_] = r GeometricProduct[e[3], ej[t,p]];

(*Row[{"j = ", j[ϕ]}]
Row[{"(e^j) = ", ej[θ,ϕ]}]*)

(*(xg[r, θ, ϕ] /. {e[1] -> e1, e[2] -> e2, e[3]->e3})- x[r, θ, ϕ] *)

xgr[r_, theta_, phi_] = D[xg[a,theta,phi],a] /. {a -> r, t->theta, p->phi};
xgt[r_, theta_, phi_] = D[xg[r,t,phi],t] /. {a -> r, t->theta, p->phi};
xgp[r_, theta_, phi_] = D[xg[r,theta,p],p] /. {a -> r, t->theta, p->phi};

{xg[r, θ, ϕ],
xgr[r, θ, ϕ],
xgt[r, θ, ϕ],
xgp[r, θ, ϕ]} // Column

{InnerProduct[xgr[r,θ,ϕ],xgt[r,θ,ϕ]],InnerProduct[xgr[r,θ,ϕ],xgp[r,θ,ϕ]],InnerProduct[xgp[r,θ,ϕ],xgt[r,θ,ϕ]],InnerProduct[xgr[r,θ,ϕ],xgr[r,θ,ϕ]],InnerProduct[xgp[r,θ,ϕ],xgp[r,θ,ϕ]],InnerProduct[xgt[r,θ,ϕ],xgt[r,θ,ϕ]]} // Simplify

OuterProduct[ xgr[r, θ, ϕ],
xgt[r, θ, ϕ],
xgp[r, θ, ϕ]]

A hybrid example to use in book

ClearAll[i,j,k,ej,x, xr, xt, xp]
i=e[1,2];
k = e[3,1];
j[phi_]=GeometricProduct[k,Cos[phi]+i Sin[phi]];
ej[t_,p_]=Cos[t]+j[p] Sin[t];
x[r_,t_,p_]=r GeometricProduct[e[3],ej[t,p]];

xr[r_,theta_,phi_]=D[x[a,theta,phi],a]/.a->r;
xt[r_,theta_,phi_]=D[x[r,t,phi],t]/.t->theta;
xp[r_,theta_,phi_]=D[x[r,theta,p],p]/.p->phi;

{x[r, θ, ϕ],
xr[r, θ, ϕ],
xt[r, θ, ϕ],
xp[r, θ, ϕ]} // Column

OuterProduct[ xr[r, θ, ϕ],
xt[r, θ, ϕ],
xp[r, θ, ϕ]]

{e1,e2,e3} = IdentityMatrix[3];
jacobean = {xr[r, θ, ϕ],
xt[r, θ, ϕ],
xp[r, θ, ϕ]} /. {e[1] -> e1, e[2] -> e2, e[3]->e3};
Det[  jacobean ] // Simplify

ClearAll[x1, x2,x3]
x1 = xr[r, θ, ϕ];
x2 = xt[r, θ, ϕ];
x3 = xp[r, θ, ϕ];
MapThread[InnerProduct,{{x1,x2,x3},{x2,x3,x1}}]// Simplify

cell output

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